Opers for Higher States of Quantum KdV Models

Abstract

We study the ODE/IM correspondence for all states of the quantum $\widehat{\mathfrak {g}}$-KdV model, where $\widehat{\mathfrak {g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak {g}$. We construct quantum $\widehat{\mathfrak {g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel (Exploring new structures and natural constructions in mathematical physics, Math. Soc. Japan, Tokyo, 2011), which are defined by fixing the singularity structure at 0 and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\widehat{\mathfrak {g}}$-KdV opers satisfy the Bethe Ansatz equations of the quantum $\widehat{\mathfrak {g}}$-KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.

1 Introduction

The purpose of the present paper is to explicitly construct and then study a class of opers, introduced by Feigin and Frenkel [20], corresponding to higher states of the quantum $\widehat{\mathfrak {g}}$-KdV model, where $\widehat{\mathfrak {g}}$ is the untwisted affinization of a simply-laced simple Lie algebra $\mathfrak {g}$. The work is the natural continuation of our previous papers in collaboration with Daniele Valeri on the ODE/IM correspondence for the ground state of the quantum $\widehat{\mathfrak {g}}$-KdV model, where we developed an effective method to construct solutions of the Bethe Ansatz equations as generalised monodromy data of affine opers [42, 43].

The quantum $\widehat{\mathfrak {g}}$-KdV model arises as the quantisation of the second Hamiltonian structure of the Drinfeld–Sokolov hierarchy [4, 6, 18]—equivalently Toda field theory [19]—as well as the continuous (conformal) limit of XXZ-like lattice models whose underlying symmetry is $U_q(\widehat{\mathfrak {g}})$ [13]. Both the lattice models and the quantum field theories carry the structure of quantum integrability (the quantum inverse scattering), so that each state is characterised by a solution of the (nested) Bethe Ansatz equations, which in turn furnishes all the physical observables of the theory.

In a ground-breaking series of papers Dorey and Tateo [15], followed by Bazhanov, Lukyanov, and Zamolodchikov [7], discovered that the solution of the Bethe Ansatz equations of the ground state of quantum $\widehat{\mathfrak {sl}_2}$-KdV (i.e. quantum KdV) admits a very simple and neat representation. Let indeed $\Psi (x,E)$ the unique subdominant solution as $x \rightarrow +\infty$ of the Schrödinger equation

$$\begin{aligned} -\psi ''(x)+(x^{2\alpha }+\frac{l(l+1)}{x^2}-E)\psi =0, \end{aligned}$$

(0.1)

with $\alpha >0$, and ${\text {Re}}l > - 1/2$. Then, $Q(E)=\lim _{x\rightarrow 0}x^{-l-1}\Psi (x,E)$ is the required solution of the Bethe Ansatz equations, with the parameters $\alpha ,l,E$ of the Schroedinger equation corresponding to the the central charge c, the vacuum parameter, and the spectral parameter of the quantum model, see [8] for the precise identification.

Such a discovery, which was thereafter known as the ODE/IM correspondence, has been generalised to many more pairs of a quantum integrable model (solvable by the Bethe Ansatz) and a linear differential operator. Examples of these generalisations, which are conjectural but supported by strong numerical evidence and deep mathematical structures, include the correspondence between all higher states of the quantum $\widehat{\mathfrak {sl}}_2$-KdV model and Schroedinger equation with ‘monster potentials’ [8], the correspondence between the ground state of massive deformations of the quantum KdV model, such as quantum Sine Gordon and quantum affine Toda theories, and the Lax operator of a dual classical theory [5, 14, 38], and the very recent discovery of the correspondence between an O(3) non-linear Sigma model and a Schroedinger operator [3]. The appearance of the Thermodynamic Bethe Ansatz in relation with BPS spectra in $\mathcal {N}=2$ Gauge theories [28], Donaldson–Thomas invariants [10], and in more general quantum mechanics equations [27, 31, 40], is also expected to be manifestations of the same phenomenon.

All these particular ODE/IM correspondences are strong evidences of the existence of an overarching ODE/IM correspondence, which can be informally stated as follows:

• Given an integrable quantum field theory, and one state of that theory, there exists a differential operator whose generalised monodromy data provide the solution of Bethe Ansatz equations of the given state.

One can make the the above conjecture much more precise for the case of the quantum $\widehat{\mathfrak {g}}$-KdV model. First of all, as discovered by Feigin and Frenkel [20], the differential operators on the ODE side of the correspondence are certain ${}^L \widehat{\mathfrak {g}}$ opers, where ${}^L \widehat{\mathfrak {g}}$ is the Langlands dual algebra of $\widehat{\mathfrak {g}}$. This implies that in the particular case under our analysis, namely when $\widehat{\mathfrak {g}}$ is the untwisted affinization of a simply laced simple Lie algebra $\mathfrak {g}$, we should consider operators with values in ${}^L \widehat{\mathfrak {g}} \cong \widehat{\mathfrak {g}}$. The ${}^L \widehat{\mathfrak {g}}$-opers proposed by Feigin and Frenkel are introduced in [20, Section 5] axiomatically, as a class of (meromorphic) ${}^L \widehat{\mathfrak {g}}$-opers satisfying certain algebraic and asymptotic assumptions. As explained below, we will make full use of these assumptions in the present paper.

The complete ODE/IM correspondence for $\widehat{\mathfrak {g}}$, with $\mathfrak {g}$ simply laced, can be then described as follows. The quantum model is defined by the choice of the central charge c and the vacuum parameter $p \in \mathfrak {h}$ of the free field representation [4, 6, 29, 33]. Every state of the Fock space is associated to a set of ${{\,\mathrm{rank}\,}}\mathfrak {g}$ entire functions $Q^{(l)}(\lambda ),l=1 \ldots {{\,\mathrm{rank}\,}}\mathfrak {g}$, of the spectral parameter $\lambda$—first introduced in [6], later generalised in [4, 29, 33], and finally settled in [25, 26] in the most general case—which solve the following Bethe Ansatz equations:

$$\begin{aligned} \prod _{j = 1}^{{{\,\mathrm{rank}\,}}\mathfrak {g}} e^{-2 i \pi \beta _jC_{\ell j}} \frac{Q^{(j)}\Big (e^{i \pi \hat{k}C_{\ell j}}\lambda ^*\Big )}{Q^{(j)} \Big (e^{- i\pi \hat{k} C_{\ell j}}\lambda ^*\Big )}=-1 \end{aligned}$$

(0.2)

for every zero $\lambda ^*$ of $Q^{(l)}(\lambda )$. In the above formula, $C_{ij}$ is the Cartan matrix of $\mathfrak {g}$, and the parameters $(\beta _1,\ldots ,\beta _{{{\,\mathrm{rank}\,}}{\mathfrak {g}}})$ and $\hat{k}$, as well as the relevant analytic properties of the functions Q’s depend on the parameters c and p—see [6, 7, 12, 42].

In order to define the ODE/IM correspondence, one needs the following data: a principal nilpotent element $f \subset \mathfrak {g}$, a Cartan decomposition $\mathfrak {g}=\mathfrak {n}_-\oplus \mathfrak {h}\oplus \mathfrak {n}_+$ such that $f=\sum _if_i \in \mathfrak {n}_-$ where $f_i$’s are the negative Chevalley generators of $\mathfrak {g}$, the dual Weyl vector $\rho ^\vee \in \mathfrak {h}$, a highest root vector $e_{\theta }$, the dual of the highest root $\theta ^\vee \in \mathfrak {h}$, an arbitrary but fixed element of the Cartan subalgebra $r\in \mathfrak {h}$, an arbitrary but fixed real number $\hat{k} \in (0,1)$, an arbitrary complex parameter $\lambda \in \mathbb {C}$, and finally a collection of pairs $\{(w_j,X(j)) \in \mathbb {C}^* \times \mathfrak {n}_+\,,\,j\in J\}$, to be determined, where $J\subset \mathbb {N}$ is a possibly empty finite set.

Given the above data, we say that a quantum$\widehat{\mathfrak {g}}$-KdV oper is an oper admitting the following representation

$$\begin{aligned} \mathcal {L}(z,\lambda )=\partial _z+\frac{r-\rho ^\vee +f}{z}+ (1+ \lambda z^{-\hat{k}})e_{\theta }+ \sum _{j \in J}\frac{-\theta ^\vee + X(j) }{z-w_j}, \end{aligned}$$

(0.3)

where in addition the regular singularities $\lbrace (w_j,X(j))\rbrace _{j \in J}$ have to be chosen so that the (0.3) has trivial monodromy at each $w_j$ for every value of $\lambda$. These further conditions ensure that the residues $-\theta ^\vee +X(j)$ belong to a $2h^{\vee }-2$ dimensional subspace of $\mathfrak {b}_+$, namely $\mathfrak {t}=\mathbb {C} \theta ^\vee \oplus [\theta ^\vee ,\mathfrak {n}_+]$, which is strictly related to the $\mathbb {Z}$-gradation on $\mathfrak {g}$ induced by the element $\theta ^\vee$, and carries a natural symplectic structure. The quantum $\widehat{\mathfrak {g}}$-KdV opers (0.3) provide an explicit realisation of the opers proposed by Feigin and Frenkel in the paper [20] (see also [26]), which was the main inspiration of the present work.

How does one attach a solution of the Bethe Ansatz equations to the above opers? The method was derived in our previous papers on the ground state oper [42, 43], which build on previous progresses by [12, 44]. Given a quantum $\widehat{\mathfrak {g}}$-KdV oper, a solution of the Bethe Ansatz equation is be constructed as follows, see Sect. 5. One considers the regular singularity at 0, and the irregular singularity at $\infty$ of (0.3). The generalised monodromy data of the oper are encoded in the connection matrix between these two singularities. This is obtained by expanding, in every fundamental representation of $\mathfrak {g}$, the subdominant solution at $\infty$ in the basis of eigensolution of the monodromy operator. These coefficients are the so-called Q functions, which satisfy the $Q\widetilde{Q}$ system, and hence the Bethe Ansatz equations.

After having introduced the Q functions, the complete Feigin-Frenkel ODE/IM conjecture [20, Section 5] for the quantum $\widehat{\mathfrak {g}}$-KdV model, with $\mathfrak {g}$ simply-laced, can be restated as follows.

Conjecture 0.1

To any state of the quantum $\widehat{\mathfrak {g}}$-KdV model there corresponds a unique quantum $\widehat{\mathfrak {g}}$-KdV oper (0.3) whose Q functions coincide with the solution of Bethe Ansatz equations of the given state. Moreover, the level $N \in \mathbb {N}$ of a state coincides with the cardinality N of the set J of additional singularities of the corresponding oper. In particular, the ground state corresponds to the case $J =\emptyset$.

In the present paper we address this correspondence, and—together with some side results which have their own independent interest in the theory of opers—we provide strong evidence of its validity by proving the following statements:

Statement 1:

The Q functions of the quantum $\widehat{\mathfrak {g}}$-KdV opers (0.3) are entire functions of $\lambda$, are invariant under Gauge transformations, and satisfy the Bethe Ansatz equations (0.2).

Statement 2:

The quantum $\widehat{\mathfrak {g}}$-KdV opers (0.3) are the most general opers which satisfy the Feigin–Frenkel axioms [20, Section 5].

Statement 3:

The parameters $\lbrace (w_j,X(j))\rbrace _{j \in J}$ of the additional singularities of the quantum $\widehat{\mathfrak {g}}$-KdV opers (0.3) are determined by a complete set of algebraic equations which are equivalent to the trivial monodromy conditions. In the particular case of a single additional singularity, and for generic values of the parameters $\hat{k}$ and r, there are ${{\,\mathrm{rank}\,}}\mathfrak {g}$ distinct quantum $\widehat{\mathfrak {g}}$- KdV opers; this number coincides with the dimension of level 1 subspace of the quantum $\widehat{\mathfrak {g}}$-KdV model.

Remark 0.2

Conjecture 0.1 does not exactly coincide with the original conjecture by Feigin and Frenkel [20, Section 5], because the explicit construction of the Q functions as the coefficients of a connection problem was still unknown, in the general case, at the time when [20] was written. Indeed, this construction was later achieved in full generality in our previous papers [42, 43], where we proved that coefficients of the connection problem satisfy a system of relations which goes under the name of $Q \widetilde{Q}$ system. The latter system was itself conjectured to hold by Dorey et al. [12] and further studied by Sun [44]. Remarkably, the same $Q \widetilde{Q}$ system was then showed by Frenkel and Hernandez [26] to hold as a universal system of relations in the commutative Grothendieck ring $K_0(\mathcal {O})$ of the category $\mathcal {O}$ of representations of the Borel subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak {g}})$, a category previously introduced by Hernandez and Jimbo in [29].

Summarising, the state-of-the-art of the ODE/IM conjecture for the quantum $\widehat{\mathfrak {g}}$-KdV model is the following (see [26] for a thorough discussion of this point). We have a putative triangular diagram whose vertices are 1) the Quantum $\widehat{\mathfrak {g}}$-KdV opers of Feigin and Frenkel, 2) the states of the Quantum $\widehat{\mathfrak {g}}$-KdV model, and 3) the solutions of the $Q\widetilde{Q}$ system with the correct analytic properties. Two arrows are now well-defined. The first, from opers to solutions of the $Q\widetilde{Q}$ system, is provided by the present work, the second, from states to solutions of the $Q\widetilde{Q}$ system, is provided in [26]. The conjecture will then be proved when a third and bijective arrow, from the states of the quantum $\widehat{\mathfrak {g}}$-KdV model to quantum opers, will be defined in such a way to make the diagram commutative.

Remark 0.3

In the $\widehat{\mathfrak {sl}}_2$ case the opers (0.3) were shown in [20] to coincide—up to a change of coordinates—to the Schrödinger operators with ‘monster potential’ studied by Bazhanov, Lukyanov, and Zamolodchikov [8]. Hence, in this case Conjecture 0.1 coincides with the one stated in [8].

Remark 0.4

A detailed description of the $\widehat{\mathfrak {sl}}_3$ case (the quantum Bousinnesq model) can now be found in [41]. That paper essentially contains the calculations of the present work specified to the $\widehat{\mathfrak {sl}}_3$ case.

Remark 0.5

The quantum $\hat{\mathfrak {g}}$-KdV opers (0.3) can be either thought of as multivalued $\mathfrak {g}$ opers, or as single valued—i.e. meromorphic—$\widehat{\mathfrak {g}}$ opers [20, 26]. Both view points will be discussed in Sect. 4.

1.1 Organization of the paper

The paper is divided in three main parts.

• 1. A preamble collecting some preliminary material, on simple and affine Lie algebras, on opers and on singularities of opers; Sects. 1–3.

• 2. The definition and analysis of quantum KdV opers, including the proof of statements 1,2 above; Sects. 4–6.

• 3. The analysis of the trivial monodromy conditions for the quantum KdV opers, including the proof of Statement 3; Sects. 7–11.

The preamble mostly consists of known material, but it contains a simple introduction to opers and their singularities—including a simple characterisation of a regular singularity, see Proposition 3.9—which may be useful to the reader. Our approach to opers is intended to be suitable to computations and to make the paper self-contained and easily accessible.

The quantum KdV opers are axiomatically defined in Sect. 4, following Feigin and Frenkel [20]. The axioms fix the singularities’ structure of the opers. They are meromorphic opers on the sphere such that 0 and $\infty$ are singularities with fixed coefficients, and all other possible singular points are regular and have trivial monodromy. To begin our analysis we drop the axiom on the trivial monodromy and deduce—after fixing an arbitrary transversal space of $\mathfrak {g}$—the canonical form of those opers which satisfy all other axioms; see Proposition 4.7. Such a canonical form does not coincide with (0.3), because in the canonical form a regular singularity is not a simple pole of the oper.

In Sect. 5, we study the generalised monodromy data of quantum KdV oper making use of their canonical form. We define the Q functions and prove that they satisfy the $Q\widetilde{Q}$ relations and thus the Bethe Ansatz equations, see Theorem 5.14. This section is based on our previous work [42], as well as on a new approach to the monodromy representation of multivalued opers.

In Sect. 6 we prove that the quantum $\widehat{\mathfrak {g}}$-KdV opers are Gauge equivalent to a unique oper of the form (0.3), see see Theorem 6.1. To this aim we introduce and study an extended Miura map. This is defined as the map that to an oper whose singularities are first order poles associates its canonical form. We prove that the extended Miura map, when appropriately restricted, is bijective.

The analysis of the trivial monodromy conditions for the quantum $\widehat{\mathfrak {g}}$-KdV opers (0.3) is divided in the five remaining sections.

In Sects. 7 and 8 we study the Lie algebra grading induced by the element $\theta ^\vee$, and we write the trivial monodromy conditions as a system of equations on the Laurent coefficients of the oper at the singular point. One of these equations is linear and is equivalent to require that the elements $-\theta ^\vee +X(j)$, for $j\in J$, belong to the $2h^{\vee }-2$ dimensional symplectic subspace $\mathfrak {t}\subset \mathfrak {b}_+$. We introduce a canonical basis for the symplectic form on $\mathfrak {t}$ and use it in Sect. 9 to derive system (9.22), which is equivalent to the trivial monodromy conditions. This is a complete system of $(2h^{\vee }-2)|J|$ algebraic equations in the $(2h^{\vee }-2)|J|$ unknowns $\lbrace (w_j,X(j)) \rbrace _{j \in J}$, which fixes the additional singularities and thus completely characterise the quantum KdV opers.

In Sect. 10 we specialise system (9.22) to the cases of the Lie algebras $A_n, n\ge 2$, $D_n, n\ge 4$, and $E_6$ (we omit to show our computations in the case $E_7,E_8$ due to their excessive length). By doing so we reduce (9.22) to a system of 2|J| algebraic equations in 2|J| unknowns. Finally, in Sect. 11 we deal with the case $\mathfrak {g}=\mathfrak {sl}_2$, which was already considered in [8, 20, 21] and requires a separate study.

2 Affine Kac–Moody Algebras

2.1 Simple Lie algebras

Let $\mathfrak {g}$ be a simply-laced simple Lie algebra of rank n, and let $h^\vee$ be the dual Coxeter number of $\mathfrak {g}$.¹ Let $\mathfrak {h}$ be a Cartan subalgebra and $\Delta \subset \mathfrak {h}^*$ be the set of roots relative to $\mathfrak {h}$. The algebra $\mathfrak {g}$ admits the roots space decomposition

$$\begin{aligned} \mathfrak {g}=\mathfrak {h}\oplus \bigoplus _{\alpha \in \Delta }\mathfrak {g}_\alpha , \end{aligned}$$

(1.1)

where $\mathfrak {g}_\alpha =\{x\in \mathfrak {g}\,|\,[h,x]=\alpha (h)x, h\in \mathfrak {h}\}$ is the root space corresponding to the root $\alpha$. Set $I=\{1,\ldots ,n\}$. Fix a set of simple roots $\Pi =\left\{ \alpha _i,i\in I\right\} \subset \Delta$, let $\Delta _+\subset \Delta$ be the corresponding set of positive roots and $\Delta _-=\Delta {\setminus } \Delta _+$ the negative roots. For $\alpha =\sum _i m_i\alpha _i\in \Delta$ define its height as ${{\,\mathrm{ht}\,}}(\alpha )=\sum _im_i\in \mathbb {Z}$. Let $\Pi ^\vee =\{\alpha _i^\vee , i\in I\}\subset \mathfrak {h}$ be simple coroots, satisfying $\langle \alpha _i^\vee ,\alpha _j\rangle =C_{ij}$ where $C=(C_{ij})_{i,j\in I}$ is the Cartan matrix of $\mathfrak {g}$. Let

$$\begin{aligned} Q=\bigoplus _{j\in I}\mathbb {Z}\alpha _j,\qquad Q^\vee =\bigoplus _{j\in I}\mathbb {Z}\alpha _j^\vee \end{aligned}$$

be respectively the root and the coroot lattice of $\mathfrak {g}$. Let $\mathcal {W}$ be the Weyl group of $\mathfrak {g}$, namely the finite group generated by the simple reflections

$$\begin{aligned} \sigma _i(\alpha _j)=\alpha _j-C_{ij}\alpha _i,\qquad i,j\in I. \end{aligned}$$

The above action on $\mathfrak {h}^*$ induces an action of $\mathcal {W}$ on $\mathfrak {h}$, with simple reflections given by

$$\begin{aligned} \sigma _i(\alpha _j^\vee )=\alpha _j^\vee -C_{ji}\alpha _i^\vee ,\qquad i,j\in I. \end{aligned}$$

Denote by $\{\omega _i, i\in I\}$ (resp. $\{\omega ^\vee _i, i\in I\}$) the fundamental weights (resp. coweights) of $\mathfrak {g}$, defined by the relations

$$\begin{aligned} \alpha _i=\sum _{j\in I}C_{ji} \omega _j,\quad \alpha _i^\vee =\sum _{j\in I}C_{ji} \omega ^\vee _j \qquad i\in I. \end{aligned}$$

Correspondingly, we denote by

$$\begin{aligned} P=\bigoplus _{j\in I}\mathbb {Z}\omega _j,\qquad P^\vee =\bigoplus _{j\in I}\mathbb {Z}\omega ^\vee _j \end{aligned}$$

(1.2)

the weight and coweight lattices of $\mathfrak {g}$. For every $\omega \in P$, we denote by $L(\omega )$ the irreducible finite dimensional highest weight $\mathfrak {g}$-module with highest weight $\omega$.

Let $\{e_{i},f_{i},i\in I\}$ be Chevalley generators of $\mathfrak {g}$, satisfying the relations

$$\begin{aligned}{}[\alpha _i^\vee ,e_{j}]=C_{ij}e_{j}, \quad [\alpha _i^\vee ,f_{j}]=-C_{ij}f_{j},\quad [e_{i},f_{j}]=\delta _{ij}\alpha _i^\vee \end{aligned}$$

(1.3)

for $i,j\in I$. Let $\mathfrak {n}_+$ (resp. $\mathfrak {n}_-$) the nilpotent subalgebra of $\mathfrak {g}$ generated by $\{e_i,i\in I\}$ (resp. $\{f_i,i\in I\}$), and recall the Cartan decomposition $\mathfrak {g}=\mathfrak {n}_-\oplus \mathfrak {h}\oplus \mathfrak {n}_+$. In addition, denote $\mathfrak {b}_+=\mathfrak {h}\oplus \mathfrak {n}_+$ the Borel subalgebra associated to the pair $(\mathfrak {g},\mathfrak {h})$. Let $\mathcal {G}$ be the adjoint group of $\mathfrak {g}$, denote by $\mathcal {B}$ the (maximal) solvable subroup of $\mathcal {G}$ whose Lie algebra is $\mathfrak {b}_+$, by $\mathcal {H}$ the abelian torus with Lie algebra $\mathfrak {h}$ and by $\mathcal {N}$ the unipotent subgroup of $\mathcal {G}$ whose Lie algebra is $\mathfrak {n}_+$. Then $\mathcal {N}$ is a normal subgroup of $\mathcal {B}$ and $\mathcal {B}=\mathcal {N}\rtimes \mathcal {H}$. Consider the exponential map $\exp :\mathfrak {n}_+\rightarrow \mathcal {N}$. Given $y\in \mathfrak {n}_+$, the adjoint action of $\exp (y)\in \mathcal {N}$ on $\mathfrak {g}$ is given by

$$\begin{aligned} \exp (y).x=x+\sum _{k\ge 1}\frac{1}{k!}({{\,\mathrm{ad}\,}}_y)^k x,\qquad \,x\in \mathfrak {g}, \end{aligned}$$

where ${{\,\mathrm{ad}\,}}_yx=[y,x]$. Define a bilinear non-degenerate symmetric form $(\cdot \vert \cdot )$ on $\mathfrak {h}$ by the equations

$$\begin{aligned} (\alpha _i^\vee \vert \alpha _j^\vee )=C_{ij}, \qquad i,j\in I \end{aligned}$$

(1.4)

and introduce the induced isomorphism $\nu :\mathfrak {h}\rightarrow \mathfrak {h}^*$ as

$$\begin{aligned} \langle h',\nu (h)\rangle =(h'\vert h), \qquad h,h'\in \mathfrak {h}. \end{aligned}$$

Note that in particular we have $\nu (\alpha _i^\vee )=\alpha _i$, $i\in I$, and the induced bilinear form $(\cdot |\cdot )$ on $\mathfrak {h}^*$ satisfies:

$$\begin{aligned} (\alpha _i\vert \alpha _j)=C_{ij}, \qquad i,j\in I \end{aligned}$$

(1.5)

follows. As proved in [32], there exists a (unique) nondegenerate invariant symmetric bilinear form $(\cdot | \cdot )$ on $\mathfrak {g}$ such that

$$\begin{aligned}&(\mathfrak {h}|\mathfrak {h})\,\,\text {is defined by (1.4)}, \end{aligned}$$

(1.6a)

$$\begin{aligned}&(\mathfrak {g}_\alpha | \mathfrak {h})=0,\quad \alpha \in \Delta , \end{aligned}$$

(1.6b)

$$\begin{aligned}&(\mathfrak {g}_\alpha | \mathfrak {g}_\beta )=0\quad \alpha ,\beta \in \Delta , \,\alpha \ne -\beta , \end{aligned}$$

(1.6c)

$$\begin{aligned}&[x,y]=(x|y)\nu ^{-1}(\alpha ), \quad x\in \mathfrak {g}_\alpha , y\in \mathfrak {g}_{-\alpha }, \alpha \in \Delta . \end{aligned}$$

(1.6d)

We will consider this bilinear form on $\mathfrak {g}$ from now on.

Let $\rho =\sum _{i\in I}\omega _i\in \mathfrak {h}^*$ be the Weyl vector, and denote

$$\begin{aligned} \rho ^\vee =\nu ^{-1}(\rho )=\sum _{i,j\in I}(C^{-1})^{ij}\alpha _j^\vee . \end{aligned}$$

The principal gradation of $\mathfrak {g}$ is defined as

$$\begin{aligned} \mathfrak {g}=\bigoplus _{i=-h^\vee +1}^{h^\vee -1}\mathfrak {g}^i, \qquad \mathfrak {g}^i=\left\{ x\in \mathfrak {g}\;\vert \; [\rho ^\vee ,x]=ix\right\} . \end{aligned}$$

(1.7)

We denote by $\pi ^j$ the projection from $\mathfrak {g}$ onto $\mathfrak {g}^j$:

$$\begin{aligned} \pi ^j: \;\mathfrak {g}\rightarrow \mathfrak {g}^j \end{aligned}$$

(1.8)

The element

$$\begin{aligned} f=\sum _{i\in I}f_i, \end{aligned}$$

(1.9)

is a principal nilpotent element. Clearly, $f\in \mathfrak {g}^{-1}$, and moreover one can prove [34] that f satisfies the following properties: ${{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_f\subseteq \mathfrak {n}_-$, $[f,\mathfrak {n}_+]\subset \mathfrak {b}_+$ and ${{\,\mathrm{ad}\,}}_{\rho ^\vee }[f,\mathfrak {n}_+]\subseteq [f,\mathfrak {n}_+]$. Since $\rho ^\vee$ is semisimple, it follows that there exists an ${{\,\mathrm{ad}\,}}_{\rho ^\vee }$-invariant subspace $\mathfrak {s}$ of $\mathfrak {b}_+$ such that

$$\begin{aligned} \mathfrak {b}_+=[f,\mathfrak {n}_+]\oplus \mathfrak {s}, \end{aligned}$$

(1.10)

and since $({{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_f)|_{\mathfrak {n}_+}=0$, then $\dim \mathfrak {s}=\dim \mathfrak {b}_+-\dim \mathfrak {n}_+=n$. The choice of $\mathfrak {s}$ is not unique, and as a possible choice of $\mathfrak {s}$ one can always take $\mathfrak {s}={{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_e$, where e is that unique element of $\mathfrak {g}$ such that $\{f,2\rho ^\vee ,e\}$ is an $\mathfrak {sl}_2$-triple. However, in this paper we do not make this specific choice, and we consider an arbitrary subspace $\mathfrak {s}$ satisfying (1.10). The affine subspace $f+\mathfrak {s}$ is known as transversal subspace; by a slight abuse of terminology, we also refer to the subspace $\mathfrak {s}$ as a transversal subspace. The space $f+\mathfrak {s}$ has the property that every regular orbit of $\mathcal {G}$ in $\mathfrak {g}$ intersects $f+\mathfrak {s}$ in one and only one point (this property justifies the terminology “transversal subspace”). In addition, for every $x\in \mathfrak {b}_+$ there exist a unique $a\in \mathcal {N}$ and a unique $s\in \mathfrak {s}$ such that $a.(f+s)=f+x$, where a. denotes the adjoint action of $\mathcal {N}$ on $\mathfrak {g}$. More precisely, the map

$$\begin{aligned} \mathcal {N}\times (f+\mathfrak {s})\rightarrow f+\mathfrak {b}_+ \end{aligned}$$

provided by the adjoint action is an isomorphism of affine varieties. Last, we introduce the concept of exponents of the Lie algebra $\mathfrak {g}$. Decomposing (1.10) with respect to the principal gradation, one obtains a set of equations of the form $\mathfrak {g}^i=[f,\mathfrak {g}^{i+1}]\oplus \mathfrak {s}^i$, for $i=0,\ldots ,h^\vee -1$, where $\mathfrak {s}^i=\mathfrak {s}\cap \mathfrak {g}^i$ and $\mathfrak {g}^{h^\vee }=\{0\}$. Since $({{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_f)|_{\mathfrak {n}_+}=0$, then $\dim \mathfrak {s}^i=\dim \mathfrak {g}^i-\dim \mathfrak {g}^{i+1}$. If $\dim \mathfrak {s}^i>0$, then i is said to be an exponent of $\mathfrak {g}$, and $\dim \mathfrak {s}^i$ is the multiplicity of the exponent i. Counting multiplicities, there are $n={{\,\mathrm{rank}\,}}\mathfrak {g}$ exponents, which we denote by $d_1,\ldots ,d_n$.

2.2 A basis for $\mathfrak {g}$

Let $\{e_i,\alpha ^\vee _i,f_i,i\in I\}\subset \mathfrak {g}$ be generators of $\mathfrak {g}$ defined as above. Following [32, § 7.8], and recalling that $\mathfrak {g}$ is simply-laced, we define a basis for $\mathfrak {g}$ as follows. For every pair of simple roots $\alpha _i,\alpha _j$, $i,j\in I$, let

$$\begin{aligned} \varepsilon _{\alpha _i,\alpha _j}= {\left\{ \begin{array}{ll} (-\,1)^{C_{ij}} &{}\quad i<j,\\ -\,1 &{}\quad i=j,\\ 1 &{}\quad i>j, \end{array}\right. } \end{aligned}$$

(1.11)

and extend this to a function $\varepsilon :Q\times Q\rightarrow \{\pm 1\}$ by bimultiplicativity:

$$\begin{aligned} \varepsilon _{\alpha +\beta ,\gamma }=\varepsilon _{\alpha ,\gamma }\varepsilon _{\beta ,\gamma },\quad \varepsilon _{\alpha ,\beta +\gamma }=\varepsilon _{\alpha ,\beta }\varepsilon _{\alpha ,\gamma }, \qquad \alpha ,\beta ,\gamma \in Q. \end{aligned}$$

(1.12)

Then, for $\alpha \in \Delta$ there exists nonzero $E_\alpha \in \mathfrak {g}$, with $E_{\alpha _i}=e_i$, $E_{-\alpha _i}=-f_i$, $i\in I$, uniquely characterized by the relations

$$\begin{aligned} {\left\{ \begin{array}{ll} {[}h,E_\alpha ]=\langle h, \alpha \rangle E_\alpha ,\qquad &{}\quad h\in \mathfrak {h},\,\alpha \in \Delta ,\\ {[}E_\alpha ,E_{-\alpha }]=-\nu ^{-1}(\alpha ),\qquad &{}\quad \alpha \in \Delta ,\\ {[}E_\alpha ,E_\beta ]=\varepsilon _{\alpha ,\beta }E_{\alpha +\beta },\qquad &{}\quad \alpha ,\beta ,\alpha +\beta \in \Delta ,\\ {[}E_\alpha ,E_\beta ]=0,\qquad &{}\quad \alpha ,\beta \in \Delta ,\,\alpha +\beta \notin \Delta \cup \{0\}. \end{array}\right. } \end{aligned}$$

(1.13)

We clearly have the root space decomposition

$$\begin{aligned} \mathfrak {g}=\mathfrak {h}\oplus \bigoplus _{\alpha \in \Delta }\mathbb {C}E_\alpha . \end{aligned}$$

(1.14)

In addition, it follows from (1.6) and (1.13) that for $\alpha , \beta \in \Delta$ we have

$$\begin{aligned} (E_\alpha |\mathfrak {h})=0,\qquad (E_\alpha |E_\beta )=-\delta _{\alpha ,-\beta }, \end{aligned}$$

(1.15)

where $(\cdot | \cdot )$ is the normalized invariant form defined in (1.6). The following result will be useful in Sect. 9.

Lemma 1.1

For every $\beta ,\gamma \in Q$, then

• (i) $\varepsilon _{\beta ,-\gamma }=\varepsilon _{\beta ,\gamma }=\varepsilon _{-\beta ,\gamma }$,

• (ii) $\varepsilon _{0,\beta }=\varepsilon _{\beta ,0}=1$,

• (iii) $\varepsilon _{\beta ,\beta }=(-1)^{\frac{1}{2}(\beta | \beta )}$,

• (iv) $\varepsilon _{\beta ,\alpha }\varepsilon _{\alpha ,\beta }=(-1)^{(\alpha |\beta )}$.

Proof

i) Let $\beta =\sum _j \beta ^j\alpha _j, \gamma =\sum _j \gamma ^j\alpha _j\in Q$. Then using (1.12) we have

$$\begin{aligned} \varepsilon _{\beta ,-\gamma }&=\prod _{i,j=1}^n(\varepsilon _{\alpha _i, \alpha _j})^{-\beta _i\gamma _j}=\prod _{j=1}^n\left( \prod _{i<j}(-1)^{-\beta _iC_{ij} \gamma _j}(-1)^{-\beta _j\gamma _j}\right) \\&=\prod _{j=1}^n\left( \prod _{i<j}(-1)^{\beta _iC_{ij}\gamma _j}(-1)^{\beta _j\gamma _j} \right) =\varepsilon _{\beta ,\gamma }. \end{aligned}$$

ii) From point i) we get $\varepsilon _{0,\beta }=\varepsilon _{\gamma -\gamma ,\beta }=\varepsilon _{\gamma ,\beta }\varepsilon _{-\gamma ,\beta }=(\varepsilon _{\gamma ,\beta })^2=1$.

iii) Let $\beta =\sum _{i=1}^n\beta ^i\alpha _i\in Q$, so that $(\beta |\beta )=\sum _{ij}\beta ^iC_{ij}\beta ^j=2\sum _{j}\left( \sum _{i<j}\beta ^iC_{ij}\beta ^j+(\beta ^j)^2\right)$, where in the last equality we used the relations $C_{ji}=C_{ij}$ and $C_{ii}=2$. Thus we have

$$\begin{aligned} \sum _{j=1}^n\left( \sum _{i<j}\beta ^iC_{ij}\beta ^j+(\beta ^j)^2\right) =\frac{1}{2}(\beta | \beta ) \end{aligned}$$

for every $\beta =\sum _i\beta ^i\alpha _i\in Q$. We now compute $\varepsilon _{\beta ,\beta }$. Using (1.11) and (1.12) we obtain

$$\begin{aligned} \varepsilon _{\beta ,\beta }&=\prod _{i,j=1}^n(\varepsilon _{\alpha _i,\alpha _j})^{\beta _i \beta _j}=\prod _{j=1}^n\left( \prod _{i<j}(-1)^{\beta _iC_{ij}\beta _j}(-1)^{(\beta _j)^2}\right) \\&=(-1)^{\sum _{j=1}^n\left( \sum _{i<j}\beta _iC_{ij}\beta _j+(\beta _j)^2\right) } =(-1)^{\frac{1}{2}(\beta | \beta )}. \end{aligned}$$

iv) Replacing in iii) $\beta$ with $\alpha +\beta$ and using (1.12) we get $\varepsilon _{\beta ,\alpha }\varepsilon _{\alpha ,\beta }=(-1)^{(\alpha |\beta )}$. $\quad \square$

2.3 Affine Kac–Moody algebras

Let $\mathfrak {g}$ be a simply-laced simple Lie algebra as above, let $\mathfrak {h}\subset \mathfrak {g}$ be a Cartan subalgebra, and fix a nondegenerate invariant bilinear form $(\cdot | \cdot )$ on $\mathfrak {g}$ as in (1.6). The untwisted affine Kac–Moody algebra $\hat{\mathfrak {g}}$ associated to the simple Lie algebra $\mathfrak {g}$ can be realized in terms of $\mathfrak {g}$ as the space

$$\begin{aligned} \hat{\mathfrak {g}}=\mathfrak {g}[\lambda ,\lambda ^{-1}]\oplus \mathbb {C}K \oplus \mathbb {C}d, \end{aligned}$$

with the commutation relations

$$\begin{aligned}{}[\lambda ^m\otimes&x\oplus a K\oplus b d,\lambda ^n\otimes y\oplus a' K\oplus b' d]\nonumber \\&=\left( \lambda ^{m+n}\otimes [x,y]-b'm \,\lambda ^m\otimes x+b\, n \,\lambda ^n\otimes y\right) \oplus m\delta _{m,-n}(x|y)K, \end{aligned}$$

where $a,b,a',b'\in \mathbb {C}$, $m,n\in \mathbb {Z}$ and $x,y\in \mathfrak {g}$. Note that K is a central element, while d acts as the derivation $\lambda \partial _\lambda$. The Cartan subalgebra of $\hat{\mathfrak {g}}$ is the finite dimensional subalgebra

$$\begin{aligned} \hat{\mathfrak {h}}=\mathfrak {h}\oplus \mathbb {C}K\oplus \mathbb {C}d. \end{aligned}$$

Let $\{e_i,f_i,i\in I\}$ be Chevalley generators of $\mathfrak {g}$, as above, and for $i\in I$ set $\hat{e}_i=1\otimes e_i$ and $\hat{f}_i=1\otimes f_i$. Moreover, let $e_\theta \in \mathfrak {g}_\theta$ (resp. $e_{-\theta }\in \mathfrak {g}_{-\theta }$) be a highest (resp. lowest) root vector for $\mathfrak {g}$ and set $\hat{e}_0=\lambda ^{-1}\otimes e_{-\theta }$, $\hat{f}_0=\lambda \otimes e_{\theta }$. Putting $\hat{I}=\{0,\ldots ,n\}$, then $\{\hat{e}_i,\hat{f}_i,i\in \hat{I}\}$ is a set of generators for $\hat{\mathfrak {g}}$. We denote by $\hat{f}$ the element $\hat{f}=\sum _{i \in \hat{I}}\hat{f}_i$.

3 Opers

In this Section we review the concept of $\mathfrak {g}$-opers and some of its basic theory. This is done in order to keep the paper as self-contained as possible and to fix the notation; consequently we follow a basic and purely algebraic approach, suitable to computations. For more details on the subject, including the geometric approach and the extension to more general groups and algebras, the reader may consult [9, 16, 23, 36] and references therein.

For any open and connected subset D of the Riemann sphere $\mathbb {P}^1$, we call $O_D$ the ring of regular functions on D, and $K_D$ the field of meromorphic functions on it. Given a $\mathbb {C}$ vector space V, we denote $V(O_D)=O_D \otimes V$ and $V(K_D)=K_D \otimes V$, namely the space of the regular/meromorphic functions on D with values in V. Opers are, locally, equivalence classes of differential operators modulo Gauge transformations. In this work we consider classes of meromorphic differential operators modulo meromorphic Gauge transformations.

The local operators under consideration belong to the classes ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D),\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$, which we define below.

Definition 2.1

Let z be a local holomorphic coordinate on $\mathbb {P}^1$ that identifies $\mathbb {P}^1$ with $\mathbb {C} \cup \lbrace \infty \rbrace$, and let $\mathcal {L}$ be a differential operator in z. We say that $\mathcal {L}$ belongs to ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ if it is of the form [9]:

$$\begin{aligned} \mathcal {L}=\partial _z+ f+ b \end{aligned}$$

(2.1)

for some $b \in \mathfrak {b}_+(K_D)$. We say that $\mathcal {L}$ belongs to $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ if it is of the form

$$\begin{aligned} \mathcal {L}=\partial _z+\sum _{i=1}^n \psi _i f_i+ b \end{aligned}$$

(2.2)

where $b \in \mathfrak {b}_+(K_D)$, and $\psi _i \in K_D {\setminus } \lbrace 0 \rbrace$, $i=1,\ldots ,n$.

The local Gauge groups we consider are $\mathcal {N}(K_D),\mathcal {H}(K_D), \mathcal {B}(K_D)$, which we introduce below together with their actions on ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D),\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$.

Definition 2.2

The unipotent Gauge group is the set

$$\begin{aligned} \mathcal {N}(K_D)=\lbrace \exp {y}, \; y \in \mathfrak {n}_+(K_D) \rbrace \end{aligned}$$

(2.3)

with the natural group structure inherited from $\mathcal {N}$. The (adjoint) action of $\mathcal {N}(K_D)$ on $\mathfrak {g}(K_D)$ is defined as

$$\begin{aligned} \exp {({{\,\mathrm{ad}\,}}y)}.g=\sum _{k\ge 0}\frac{1}{k!}({{\,\mathrm{ad}\,}}_y)^k \, g, \qquad y \in \mathfrak {n}_+(K_D),\, g \in \mathfrak {g}(K_D). \end{aligned}$$

The adjoint action of $\mathcal {N}(K_D)$ on $\partial _z$ is expressed by Dynkin’s formula

$$\begin{aligned} \exp {({{\,\mathrm{ad}\,}}y)}.\partial _z=\partial _z-\sum _{k\ge 0}\frac{1}{(k+1)!}({{\,\mathrm{ad}\,}}_{y})^k\frac{d y}{dz}, \qquad y \in \mathfrak {n}_+(K_D), \end{aligned}$$

(2.4)

which is equivalent to $N.\partial _z=\partial _z-\frac{dN}{dz}N^{-1}$, for $N=\exp {y}$.

Remark 2.3

Let us extend the algebra structure of $\mathfrak {n}_+(K_D)$ to the space $\mathfrak {n}_+(K_D) \oplus \mathbb {C} \partial _z$ by the formula $[\partial _z,y]=\frac{dy}{dz}$. Then formula (2.4) for the action of $\exp {y}$ on $\partial _z$ coincides with the adjoint action according to the bracket of the extended algebra. Indeed,

$$\begin{aligned} \sum _{l=0}^{\infty }\frac{1}{l!}({{\,\mathrm{ad}\,}}_y)^l\partial _z= \partial _z+\sum _{l=1}^{\infty }\frac{1}{l!}{{\,\mathrm{ad}\,}}^{l-1}_{y}[y,\partial _z]= \partial _z-\sum _{k=0}^{\infty }\frac{1}{(k+1)!}{{\,\mathrm{ad}\,}}^k_y \frac{dy}{dz}. \end{aligned}$$

Definition 2.4

We denote $\mathcal {H}(K_D)$ the abelian mutiplicative group generated by elements of the form $\varphi ^{\lambda }$ for $\varphi \in K_D{\setminus } \lbrace 0 \rbrace$ and $\lambda \in P^{\vee }$, the co-weight lattice (1.2). Since ${{\,\mathrm{rank}\,}}P^\vee =n$ then $\mathcal {H}(K_D)$ is isomorphic to $\left( K_D{\setminus } \lbrace 0 \rbrace \right) ^n$. The (adjoint) action of $\mathcal {H}(K_D)$ on $\mathfrak {g}(K_D)$ is given by means of the root space decomposition (1.1): if $g=g_0+\sum _{\alpha \in \Delta } g_{\alpha }\in \mathfrak {g}(K_D)$, with $g_0\in \mathfrak {h}(K_D)$ and $g_{\alpha } \in \mathfrak {g}_{\alpha }(K_D)$ then

$$\begin{aligned} \varphi ^{{{\,\mathrm{ad}\,}}\lambda }.g=g_0+ \sum _{\alpha \in \Delta } \varphi ^{\alpha (\lambda )}g_{\alpha }. \end{aligned}$$

The adjoint action of $\mathcal {H}(K_D)$ on the operator $\partial _z$ is given by

$$\begin{aligned} \varphi ^{{{\,\mathrm{ad}\,}}\lambda }. \partial _z=\partial _z -\frac{\varphi '}{\varphi }\lambda . \end{aligned}$$

Finally, the action of $\mathcal {H}(K_D)$ on $\mathfrak {n}_+(K_D)$ induces an action on $\mathcal {N}(K_D)$ as follows

$$\begin{aligned} \varphi ^{{{\,\mathrm{ad}\,}}\lambda }.\exp {y}=\exp {\big (\varphi ^{ {{\,\mathrm{ad}\,}}\lambda }. y\big )},\qquad y \in \mathfrak {n}_+(K_D), \, \varphi \in K_D{\setminus } \{0\}. \end{aligned}$$

Definition 2.5

Given the above action of $\mathcal {H}(K_D)$ on $\mathcal {N}(K_D)$, we define $\mathcal {B}(K_D)=\mathcal {H}(K_D) \rtimes \mathcal {N}(K_D)$ as the semidirect product induced by it, namely:

$$\begin{aligned} \big (\exp {n},\varphi ^{ \lambda } \big ) \big (\exp {m},\psi ^{\mu }\big )= \big (\exp {({{\,\mathrm{ad}\,}}n)}.\exp {(\varphi ^{ {{\,\mathrm{ad}\,}}\lambda } m)},\varphi ^{ \lambda }\psi ^{ \mu }), \end{aligned}$$

for $n,m\in \mathfrak {n}_+(K_D)$ and $\varphi ^\lambda ,\psi ^\mu \in \mathcal {H}(K_D)$.

Summing up the previous definitions, we can explicitly write the action of $\mathcal {N}(K_D)$ on $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ (and in particular on ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$) as

$$\begin{aligned} \exp {({{\,\mathrm{ad}\,}}y)}. \big ( \partial _z+\sum _{i}\psi _if_i+b\big )=\,&\partial _z+ \sum _{k\ge 0} \frac{1}{k!}({{\,\mathrm{ad}\,}}_y)^k \big (\sum _i \psi _i f_i+b\big ) \nonumber \\&-\sum _{k\ge 0}\frac{1}{(k+1)!}({{\,\mathrm{ad}\,}}_y)^k\frac{d y}{dz}, \end{aligned}$$

(2.5)

with $y\in \mathfrak {n}_+(K_D)$. It is immediate to prove (using the principal gradation (1.7)) that the action (2.5) is free. Similarly, the action of $\mathcal {H}(K_D)$, and thus of $\mathcal {B}(K_D)$, on $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ is given by:

$$\begin{aligned} \varphi ^{{{\,\mathrm{ad}\,}}\lambda }.\big ( \partial _z+\sum _{i}\psi _if_i+b\big )&= \partial _z+ \sum _{i}\varphi ^{-\alpha _i(\lambda )}\psi _if_i+\nonumber \\&\quad - \frac{\varphi '}{\varphi }\lambda +b_0+ \sum _{\alpha \in \Delta _+}\varphi ^{\alpha (\lambda )}b_{\alpha }, \end{aligned}$$

(2.6)

where $b=b_0+\sum _{\alpha \in \Delta _+}b_\alpha$, with $b_0\in \mathfrak {h}(K_D)$ and $b_\alpha \in \mathfrak {g}_\alpha (K_D)$. Formula (2.6) has two immediate consequences:

• 1. The only element in $\mathcal {H}(K_D)$ that leaves the set ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ invariant is the identity

• 2. For any choice of the functions $\psi _i$, $i\in I$, there is a unique element in $\mathcal {H}(K_D)$ that maps $\mathcal {L} \in \widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ to an operator in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$; explicitly this is $\prod _j \psi _j(z)^{\omega _j^\vee }$ where $\omega _j^\vee$, $j\in I$, are the fundamental co-weights.

It follows from the above that there is a bijection between the sets of equivalence classes ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)/\mathcal {N}(K_D)$ and $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)/\mathcal {B}(K_D)$.

Definition 2.6

Let D be an open, connected and simply-connected subset of $\mathbb {P}^1$. The space of opers ${{\,\mathrm{Op}\,}}_{\mathfrak {g}}(D)$ is defined as ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)/\mathcal {N}(K_D)\cong \widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)/\mathcal {B}(K_D)$. We denote by $[\mathcal {L}]$ the equivalence class (i.e. the oper) of the operator $\mathcal {L}$.

Fixed a transversal space $f+\mathfrak {s}$, then each equivalence class of operators in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ admits a unique representative of the form $\partial _z+f+s$, with $s\in \mathfrak {s}(K_D)$. The space of opers on a domain D of the Riemann sphere was essentially described in the holomorphic case—namely considering holomorphic operators and holomorphic Gauge transforms—by Drinfeld and Sokolov (see [16, Proposition 6.1]). Since in the present paper we consider meromorphic operators and meromorphic Gauge transforms, we need a slightly extended version of that proposition. Hence, we review its proof too.

Definition 2.7

Let $f+\mathfrak {s}$ a transversal space. Given the splitting $\mathfrak {b}_+=[f,\mathfrak {n}_+]\oplus \mathfrak {s}$, we denote $\Pi _{f}:\mathfrak {b}_+\rightarrow [f,\mathfrak {n}_+]$ and $\Pi _{\mathfrak {s}}:\mathfrak {b}_+\rightarrow \mathfrak {s}$ the respective projections.

Proposition 2.8

(cf. Proposition 6.1 in [16]) Let $f+\mathfrak {s}$ be a transversal space. For every meromorphic differential operator $\mathcal {L}=\partial _z+f+b \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$, there exists a unique meromorphic function $s \in \mathfrak {s}(K_D)$ and a unique Gauge transform $N\in \mathcal {N}(K_D)$ such that $N.\mathcal {L}=\partial _z+f+s$. Furthermore, the set of singular points of s is a subset of the set of singular points of b.

Proof

We first prove the existence of the pair N, s, and then its uniqueness. We construct—by induction with respect to the principal gradation—the pair N, s as $N=N_{h^\vee -1}\cdots N_1$, with $N_i=\exp y^i$ and $y^i\in \mathfrak {g}^i(K_D)$ and $s=\sum _{i=1}^{h^\vee -1}s^i$, with $s^i\in \mathfrak {s}^i(K_D)$. Let $\mathcal {L}=\partial _z+f+b\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$, and let $b=\sum _{i=0}^{h^\vee -1}b^i$, with $b^i\in \mathfrak {g}^i(K_D)$. Introduce $N_1=\exp y^1$ with $y^1\in \mathfrak {g}^1(K_D)$ and set $\mathcal {L}_1=N_1\mathcal {L}$. Due to (2.5) then $\mathcal {L}_1=\partial _z+f+b^0+[y^1,f]+\sum _{i=1}^{h^\vee -1}\bar{b}^i$, for certain $\bar{b}^i\in \mathfrak {g}^1(K_D)$. Note that $b^0\in \mathfrak {h}\subseteq [f,\mathfrak {n}_+]$, and since ${{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_f$ is trivial on $\mathfrak {n}_+$, we take $y^1$ to be the unique solution of the equation $b^0+[y^1,f]=0$, so that $\mathcal {L}_1$ takes the form $\mathcal {L}_1=\partial _z+f+\sum _{i=1}^{h^\vee -1}\bar{b}^i$. Note that by construction $y^1$ has at most the same singularities of $b^0$, which is an element of $\mathcal {L}$. Since $\mathcal {L}_1$ is generated from $\mathcal {L}$ by the iterated adjoint action of $y^1$, then the set of singular points of $\mathcal {L}_1$ is contained in the set of singular point of $\mathcal {L}$. Now fix $j>1$ and assume we found elements $N^{l}=\exp {n^l}$, $l=1,\ldots ,j-1$ with $n^l \in \mathfrak {g}^l(K_{D})$, as well as $s^l\in \mathfrak {s}^l(K_D)$, $l=1,\ldots ,j-2$ such that

$$\begin{aligned} \mathcal {L}_{j-1}:=N_{j-1}\cdots N_1.\mathcal {L}=\partial _z+f+\sum _{l=1}^{j-2}s^l+\sum _{l=j-1}^{h^\vee -1}c^l, \end{aligned}$$

for some $c^l\in \mathfrak {g}^l(K_D)$. Assume moreover that the set of singular points of $\mathcal {L}_{j-1}$ is contained in $S_b$. Introduce $N_j=\exp y^j$ with $y^j\in \mathfrak {g}^j(K_D)$ and set $\mathcal {L}_j=N_j\mathcal {L}_{j-1}$. Using (2.5) we obtain

$$\begin{aligned} \mathcal {L}_j=\partial _z+f+\sum _{l=1}^{j-2} s^l+ [y^j, f]+ \bar{c}^{j-1}+\sum _{l=j}^{h^\vee -1} \bar{c}^l, \end{aligned}$$

for some $\bar{c}^l\in \mathfrak {g}^l(K_D)$. We are interested in the term $[y^j, f]+ \bar{c}^{j-1}$. Recalling the projection operators given in Definition 2.7, then we define $s^j=\Pi _{\mathfrak {s}}(\bar{c}^{j-1})\in \mathfrak {s}^j(K_D)$, and we take $y^j$ to be the unique solution of the equation $[y^j,f]+\bar{c}^{j-1}=0$. Such a solution exists and is unique since $({{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_f)|_{\mathfrak {n}_+}=0$. Then, $\mathcal {L}_j$ takes the form

$$\begin{aligned} \mathcal {L}_j=\partial _z+\sum _{l=1}^{j-1} s^l+\sum _{l=j}^{h^\vee -1} \bar{c}^l. \end{aligned}$$

By construction, $y^j$ has at most the same singularities of $\bar{c}^{j-1}$, which is an element of $\mathfrak {L}_{j-1}$. Since $\mathcal {L}_j$ is generated by the action of $y^j$ on $\mathcal {L}_{j-1}$, the singular locus of $\mathcal {L}_j$ is a subset of the singular locus of $\mathcal {L}_{j-1}$. Iterating the above procedure, one obtains elements $N=N_{h^\vee -1}\ldots N_1$, with $N_j=\exp y^j$ and $y^j\in \mathfrak {g}^j(K_D)$, and $s=\sum _{i=1}^{h^\vee -1}s^i$ with $s^i\in \mathfrak {s}^i(K_D)$, so that $N.\mathcal {L}=\partial _z+f+s$, and the set of singular points of $N.\mathcal {L}$—namely the singular points of s—is contained in the set of singular points of $\mathcal {L}$, namely the singular points of b. Note incidentally that $s^i=0$ if i is not an exponent of $\mathfrak {g}$.

The pair (N, s) constructed above is unique, because the action of $\mathcal {N}(K_D)$ on ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ is free, and if two operators of the form $\partial _z+f+s$, $\partial _z+f+s'$, with $s, s' \in \mathfrak {s}(K_D)$ are gauge equivalent then $s=s'$. We prove the latter statement as follows. Let the two operators be Gauge equivalent, by the transformation $M=\exp {m}, m \in \mathfrak {n}_+(K_D)$, then $m=0$. Indeed suppose $m\ne 0$ and let $m^i \ne 0, m_i \in \mathfrak {g}^i(K_D)$ be the non-trivial term of m with lowest principal degree. Then $\Pi _f(\exp {m}(\partial _z+f+s)-\partial _z-f)$ has a non-trivial term of degree $i-1$, namely $[m^i,f]$, hence it is not zero. $\quad \square$

As a corollary we have the following characterisation of opers

Proposition 2.9

Let D be an open, connected, and simply connected subset of $\mathbb {C}$. After fixing a transversal space $\mathfrak {s}$, the set ${{\,\mathrm{Op}\,}}_{\mathfrak {g}}(D)$ can be identified with $\mathfrak {s}(K_D)$.

Definition 2.10

We say that an operator $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ is in canonical form if it is of the form $\mathcal {L}=\partial _z+f+s$ with $s \in \mathfrak {s}(K_D)$. We also say that $\mathcal {L}_s=\partial _z+f+s$ with $s\in \mathfrak {s}(K_D)$ is the canonical form of any element of $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ Gauge-equivalent to it.

3.1 Change of coordinates: global theory

The global theory of opers was developed in [9, Section 3]; see also [23, Chapter 4] or [36, Section 6.1], which we follow. Here we just address the simplest aspect of the global theory, that is the coordinate transformation laws of opers. Let $\Sigma$ be a Riemann surface (we will be interested here in the case $\Sigma =\mathbb {C}\mathbb {P}^1$ only), and D a chart on $\Sigma$ with coordinate z. Let $\mathcal {L}\in \widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ be of the form

$$\begin{aligned} \mathcal {L}=\partial _z+\sum _i \psi _i(z) f_i+b(z). \end{aligned}$$

If $z=\varphi (x)$ is a local change of coordinates we define the transformed operator of $\mathcal {L}$ as

$$\begin{aligned} \mathcal {L}^{\varphi }=\partial _x+\varphi '(x)\big (\sum _{i}\psi _i(\varphi (x))f_i+b(\varphi (x)) \big ). \end{aligned}$$

(2.7)

thus considering $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$ as a space of meromorphic connections on the trivial bundle $D \times \mathfrak {g} \rightarrow D$. We note that if $\widetilde{\mathcal {L}}=\exp {n(z)}.\mathcal {L}$ then $(\widetilde{\mathcal {L}})^{\varphi }=\exp {n\big (\varphi (x)\big )}. \mathcal {L}^{\varphi }$, which implies that the transformation law is compatible with quotienting by the Gauge groups.

Hence, one can define a sheaf of (meromorphic) opers ${{\,\mathrm{Op}\,}}_{\mathfrak {g}}(\Sigma )$ on the Riemann surface $\Sigma$ as follows. For A a set, let $\{U_{\alpha }\}_{\alpha \in A}$ be an open covering of charts in $\Sigma$, with transition functions $\varphi _{\alpha ,\beta }$ whenever $U_\alpha \cap U_\beta \ne \emptyset$, and let $[\mathcal {L}_{\alpha }] \in {{\,\mathrm{Op}\,}}_{\mathfrak {g}}(U_{\alpha })=\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_{U_\alpha })/\mathcal {B}(K_{U_\alpha })$ a collection of local sections of opers. An oper on $\Sigma$, namely an element on ${{\,\mathrm{Op}\,}}_{\mathfrak {g}}(\Sigma )$, is then defined as $\{[\mathcal {L}_{\alpha }], \alpha \in A\}$, with the additional requirement that on each non-empty intersection $U_{\alpha }\cap U_{\beta }$ we have that $[\mathcal {L}_{\alpha }]=[\mathcal {L}_{\beta }^{\varphi _{\alpha ,\beta }}]\in {{\,\mathrm{Op}\,}}_{\mathfrak {g}}(U_{\alpha }\cap U_{\beta })$, where $\mathcal {L}_{\beta }^{\varphi _{\alpha ,\beta }}$ is given by formula (2.7), with $\mathcal {L}=\mathcal {L}_{\beta }$ and $\varphi =\varphi _{\alpha ,\beta }$.

Remark 2.11

For a given $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$, in general $\mathcal {L}^{\varphi }$ belongs to $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_{\varphi ^{-1}(D)})$ but not to ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\varphi ^{-1}(D)})$. It is convenient to define, for any $\varphi$, an element $\widetilde{\mathcal {L}} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\varphi ^{-1}(D)})$ equivalent to $\mathcal {L}^{\varphi }$. We make the following choice:

$$\begin{aligned} \widetilde{\mathcal {L}}= \varphi '(x)^{\rho ^\vee }\mathcal {L}^{\varphi }=\partial _x+f-\frac{\varphi ^{\prime \prime }(x)}{\varphi ^\prime (x)}\rho ^\vee +\sum _{i=0}^{h^\vee -1}\left( \varphi ^\prime (x)\right) ^{i+1} b^i(\varphi (x)) \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\varphi ^{-1}(D)}), \end{aligned}$$

(2.8)

where we have decomposed $b(z)=\sum _{i=0}^{h^{\vee }-1}b^i(z)$ according to the principal gradation. Hence $[\mathcal {L}^{\varphi }]=[\widetilde{\mathcal {L}}] \in {{\,\mathrm{Op}\,}}_{\mathfrak {g}}(K_{\varphi ^{-1}(D)})$.

In the present work, we deal with meromorphic opers on the sphere ${{\,\mathrm{Op}\,}}_{\mathfrak {g}}(\mathbb {P}^1)$, whose space of global sections we characterise here. We cover $\mathbb {P}^1$ by two charts $U_{0}, U_{\infty }$ with coordinates z, x and transition function $z=\frac{1}{x}$. Suppose that we are given an operator $\partial _z+f+b(z)$ in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(U_0)$ and one operator $\partial _x+f+\tilde{b}(x)$ in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(U_{\infty })$. These are local sections of the same global oper if and only if $\partial _x+f+\tilde{b}(x)$ is gauge equivalent to the following operator

$$\begin{aligned} \mathcal {L}=\partial _x+f-\frac{2\rho ^\vee }{x}+\sum _{i=0}^{h^\vee -1}\left( \frac{-1}{x^2}\right) ^{i+1} b^i(\frac{1}{x}). \end{aligned}$$

(2.9)

Hence the operator $\partial _z+f+b(z)$, defined locally on $U_0$, can be extended to a global meromorphic oper on the sphere if and only if b(z) admits a meromorphic continuation at infinity, i.e. b(z) is a rational function. From this, it follows immediately that the space of global sections of meromorphic opers on the Riemann sphere ${{\,\mathrm{Op}\,}}_{\mathfrak {g}}(K_{\mathbb {P}^1})$ is isomorphic to $\mathfrak {s}(K_{\mathbb {P}^1})$: an oper on the sphere is defined by the choice of a transversal space and of n arbitrary rational functions.

4 Singularities of Opers

In this section we address the theory of regular and irregular singularities for differential operators in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ as well as for opers in $[\mathcal {L}]\in {{\,\mathrm{Op}\,}}_{\mathfrak {g}}(K_D)$. This theory was already addressed in the opers literature, see [9, 23, 24] among others. Here we both review known facts and include results from the literature of singularities of connections, in particular from [1, 2]. We will always point out below whenever our nomenclature deviates from the one commonly used in the opers literature.

Since we are both interested in single operators and in equivalence classes, we need to distinguish properties which are Gauge invariant and properties which are not. For example a singular point for an operator may be a regular point for a Gauge equivalent one, because we allow singular (meromorphic) Gauge transformations. Hence we start with the following

Definition 3.1

We say that a pole w of $b \in \mathfrak {b}^+(K_D)$ is a removable singularity of the differential operator $\mathcal {L}=\partial _z+f+b\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ if there exists $N \in \mathcal {N}(K_D)$ such that $N.\mathcal {L}$ is regular at w.²

The theory of singular points begins with a dichotomy, the distinction between regular and irregular singular point. In order to define it, we need to introduce the concept of algebraic behaviour.

Definition 3.2

Let $\mathbb {D}$ be the punctured disc of centre w. We say that a, possibly multivalued, function $f: \mathbb {D} \rightarrow \mathbb {C}^n, n \ge 0$ has algebraic behaviour at $z=w$ if, for any fixed closed sector S of opening less than $2 \pi$, the following estimate holds $|f(z)|=o(|z-w|^{\alpha })$ for some $\alpha \in \mathbb {R}$.

Definition 3.3

A singularity $w \in D$ of the operator $\mathcal {L}=\partial _z+f+b\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ is called regular if the following property holds for every finite dimensional module V of $\mathfrak {g}$: every local solution $y: \mathbb {C} \rightarrow V$ of the linear equation $\mathcal {L}.y=0$ has algebraic behaviour at w. A singular point that is not regular is named irregular.

The above definition is clearly Gauge invariant. It is in practice a notoriously difficult task the one of establishing whether the singularity of a connection is regular or not, see e.g. [2, Chapter 5]. However, this problem can be easily solved for the class of operators belonging to $\widetilde{{{\,\mathrm{op}\,}}}_{\mathfrak {g}}(K_D)$, as we show in Proposition 3.9 below. To this aim we start by introducing the concept of slope of the singular point [11, 24].³

Definition 3.4

Let $\mathcal {L}=\partial _z+f+b\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$, and decompose $b=\sum _{i=0}^{h^\vee -1} b^{i}$ according to the principal gradation of $\mathfrak {g}$, that is, $b^i\in \mathfrak {g}^i(K_D)$. Assume $w\ne \infty$ is a pole of b, and let $b^i=\bar{b}^i (z-w)^{-\delta _i}+O((z-w)^{-\delta _i+1})$ as $z\rightarrow w$, for some $\bar{b}^i \in \mathfrak {g}^i$ and $\delta _i \in \mathbb {Z}$. We denote

$$\begin{aligned} \mu =\max \lbrace 1,\max _{i}\frac{\delta _i}{i+1}\rbrace ,\qquad \bar{b}=\sum _{\frac{\delta _i}{i+1}=\mu }\bar{b}^i. \end{aligned}$$

(3.1)

The slope of $\mathcal {L}$ at w is defined as the rational number $\mu$, and the principal coefficient of the singularity is defined as $f-\rho ^\vee +\bar{b}$ if $\mu =1$, and as $f+\bar{b}$ if $\mu >1$.

If $w=\infty$, let $b^i(z)=\bar{b}^iz^{\delta _i}+O(z^{\delta _i-1})$ as $z\rightarrow \infty$, for some $\bar{b}^i \in \mathfrak {g}^i$ and $\delta _i \in \mathbb {Z}$. We denote

$$\begin{aligned} \mu _{\infty }=\max \lbrace -1, \max _{i}\frac{\delta _i}{i+1} \rbrace , \qquad \bar{b}=\sum _{\frac{\delta _i}{i+1}=\mu _{\infty }}\bar{b}^i. \end{aligned}$$

(3.2)

The slope of $\mathcal {L}$ at $\infty$ is defined as the rational number $\mu _{\infty }$, and the principal coefficient of the singularity is defined as $f-\rho ^\vee +\bar{b}$ if $\mu _{\infty }=-1$, and as $f+\bar{b}$ if $\mu _{\infty }>-1$.

Definition 3.5

Let $\mathcal {L}=\partial _z+f+b\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$, and let $w \in D\subset \mathbb {P}^1$ be a pole of $b\in \mathfrak {b}_+(K_D)$ or the point $w=\infty$. Then, $w\ne \infty$ is said to be a Fuchsian singular point of $\mathcal {L}$ if its slope $\mu$ is equal to 1. Equivalently, $w\ne \infty$ is Fuchsian if $(z-w)^{i+1}b^{i}(z)$ is analytic at w for all i. The point $w=\infty$ is said to be a Fuchsian singular point of $\mathcal {L}$ if its slope $\mu _{\infty }$ is equal to $-1$.

In the above definitions the point $w=\infty$ is treated separately. Note, however, that in the coordinate $x=z^{-1}$, the operator $\mathcal {L}$ is given by formula (2.9), whose slope $\mu$ at $x=0$ coincides with $\mu _{\infty }+2$. Our definitions are therefore consistent under the change of coordinates.

Remark 3.6

The authors of [9] use a different nomenclature: Equivalence classes of opers with a Fuchsian singularity, with respect to the action of Gauge transformations regular at w, are called $(\le 1)$-singular opers. In [23], the latter opers are called opers with regular singularity. Since we allow for meromorphic Gauge transformations and we do not fix a-priori a Gauge, we need to distinguish between the notion of a regular singular point and of a Fuchsian singular point. We have chosen the latter name because, in the case of $\mathfrak {sl}_n$ opers, the definition of a Fuchsian singularity coincides with the usual one in the theory of scalar ODEs, see Corollary 3.11 below.

The reason for the previous definition comes form the following observation. Let $w\ne \infty$ be a singularity of $\mathcal {L}$, with slope $\mu$ and principal coefficient $\bar{b}$ as in (3.1). Introduce a branch of $(z-w)^{\mu }$, and let $\widehat{K}_D$ be the finite extension of $K_D$ obtained by adjoining $(z-w)^{\mu }$. The Gauge transform $(z-w)^{\mu {{\,\mathrm{ad}\,}}\rho ^\vee } \in \mathcal {H}(\widehat{K}_D)$ has the following action on $\mathcal {L}$:

$$\begin{aligned} (z-w)^{\mu {{\,\mathrm{ad}\,}}\rho ^\vee }\mathcal {L}=\partial _z-\frac{ \mu \rho ^\vee }{z-w}+ \frac{f+\bar{b}}{(z-w)^{\mu }}+o((z-w)^{-\mu }), \text{ as } z \rightarrow w. \end{aligned}$$

(3.3)

From the above computation it follows that if the singularity w is Fuchsian, then $\mathcal {L}$ is locally Gauge equivalent to a differential operator with a first order pole. Its associated connection is then Fuchsian (in the sense of connections) at w, hence the singularity is regular. The same is true for the point at infinity; indeed, we have

$$\begin{aligned} z^{-\mu _{\infty } {{\,\mathrm{ad}\,}}\rho ^\vee }\mathcal {L}=\partial _z+\frac{ \mu _{\infty } \rho ^\vee }{z}+ \big (f+\bar{b}\big )z^{\mu _{\infty }}+o(z^{\mu _{\infty }}), \text{ as } z \rightarrow \infty . \end{aligned}$$

where $\mu _{\infty }$ and $\bar{b}$ are given in (3.2) (recall that infinity is a Fuchsian singularity of a linear connection on $\mathbb {P}^1$ if it is a simple zero).

We can also establish a partial converse of the above statements in case the function b takes values in any subset of $f+\mathfrak {b}$ whose only nilpotent is f; this is proved in the lemma below together with other results that will be used in the sequel. Although original within the present setting, the results below are based on well-known properties on the structure of singularities of linear equations. The appropriate references are provided in the proof.

Lemma 3.7

Let $\mathcal {L}=\partial _x+f+b \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ and w a pole of b or the point $w=\infty$.

• 1. If w is a Fuchsian singular point then it is a regular singularity.

• 2. Let $\mathfrak {m} \subset \mathfrak {b}_+$ a vector subspace of $\mathfrak {b}_+$ that satisfies the following property: $f+m$ with $m \in \mathfrak {m}$ is nilpotent if and only if $m=0$. Let $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ be of the form $\partial _z+f+m$, with $m \in \mathfrak {m}(K_D)$. The singularity at w is regular if and only if it is Fuchsian.

• 3. If $\mathcal {L},\mathcal {L}' \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ are two Gauge equivalent operators with a Fuchsian singularity at w, then the principal coefficient of $\mathcal {L}$ at w is conjugated in $\mathcal {N}$ to the principal coefficient of $\mathcal {L}'$ at w.

• 4. Let $\rho :\mathfrak {g} \rightarrow End(V)$ be a non-trivial irreducible representation of $\mathfrak {g}$ such that all local solutions of the equation $\mathcal {L}\psi =0$ have algebraic growth. Then w is a regular singularity.

Proof

We can assume $w=0$.

• 1. Due to (3.3), if 0 is Fuchsian then $z^{\rho ^\vee }\mathcal {L}$ has a simple pole at 0. Hence in every representation every solution has algebraic growth, hence $z=0$ is regular.

• 2. Because of (1), we just need to prove that not-Fuchsian implies irregular. Suppose then that $z=0$ is not Fuchsian, so that $\mu >1$. Due to (3.3), applying the gauge transform $z^{\mu {{\,\mathrm{ad}\,}}\rho ^\vee }$ to $\mathcal {L}$ then we get

  $$\begin{aligned} \partial _z+z^{-\mu } \big ( f+ \bar{m}\big )+ o(z^{-\mu }). \end{aligned}$$

  (3.4)

  where $\bar{m}\in \mathfrak {m}$ is non-zero since $\mu >1$ (cf. Definition 3.4). Since $\bar{m}\ne 0$, by hypothesis on $\mathfrak {m}$ we have that the principal coefficient $f+\bar{m}$ is not nilpotent. It follows that, fixed the adjoint representation, the operator (3.4) has a singularity with Poincaré rank greater than 1 and with a not-nilpotent principal coefficient, hence the singularity is irregular. See e.g. [45].

• 3. The proof is deferred to Lemma 6.6(2).

• 4. After [2, Theorem 5.2], the operator $\mathcal {L}$ can be brought—by means of a meromorphic Gauge transformation—into one of the two following forms:

  • (i) $\partial _z+\frac{A}{z}+O(z^{-1+\varepsilon })$, with $A\in \mathfrak {g}$,

  • (ii) $\partial _z+\frac{B}{z^{r}}+O(z^{-r+\varepsilon })$ with $r \in \mathbb {Q}$, $r>1$, and $B\in \mathfrak {g}$ is not nilpotent.

  Let V be a non-trivial $\mathfrak {g}$-module. Assuming we are in case (ii), then the matrix operator representing $\mathcal {L}$ in V has a singularity at 0 of order $r>1$ with a non nilpotent coefficient. It follows that in this case there exists at least one solution with non-algebraic behaviour. Then, all solutions (in any representation) are regular at 0 if and only if $\mathcal {L}$ can be brought to the form (i). But this implies that 0 is a regular singularity. $\quad \square$

As shown in Lemma 3.7, the subspaces of $f+\mathfrak {b}^+$ such that f is the only nilpotent play an important role in the study of regular singularities for operators in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$. Clearly $f+\mathfrak {h}$ is one example of such subspaces. Other examples are the transversal spaces $f+\mathfrak {s}$:

Proposition 3.8

(Kostant). Let $f+\mathfrak {s}$ be a transversal space. Then f is the only nilpotent element in $f+\mathfrak {s}$.

Proof

The proof of Kostant [35] follows the steps: Any transversal space is in bijection with regular orbits. The only nilpotent regular orbit is the principal nilpotent orbit. Since f is principal nilpotent, it is the only nilpotent element in the transversal space. $\quad \square$

Combining the above lemma and proposition, we deduce that if an operator is in its canonical form then a singularity is regular if and only if it is Fuchsian.

Proposition 3.9

Fix a transversal space $f+\mathfrak {s}$ and let $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ be in canonical form $\mathcal {L}=\partial _z+ f+ s$, $s \in \mathfrak {s}(K_D)$. A point $w \in \mathbb {C}$ is a regular singular point of $\mathcal {L}$ if and only if it is a not-removable Fuchsian singular point.

Proof

Because of Proposition 3.8, $\mathfrak {s}$ satisfies the hypothesis of Lemma 3.7(2), hence for an oper in canonical form a singular point is regular if and only if is Fuchsian. Moreover, since the singular locus of an operator in canonical form is a subset of the singular locus of any operator Gauge equivalent to it, then a singularity of an operator in canonical form cannot be removed. $\quad \square$

The above proposition has a two immediate corollaries. The first is a characterisation of regular singularities for $\mathfrak {g}$ opers.

Corollary 3.10

Fix a transversal space $f+\mathfrak {s}$, let $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ and $\mathcal {L}_{\mathfrak {s}}$ be its canonical form. All regular points of $\mathcal {L}$ are regular points of $\mathcal {L}_{\mathfrak {s}}$, and all regular singular points of $\mathcal {L}$ are either regular points or not-removable Fuchsian singular points of $\mathcal {L}_{\mathfrak {s}}$.

Proof

Let $N\in \mathcal {N}(K_D)$ be the Gauge transformation mapping $\mathcal {L}$ to its canonical form, namely $\mathcal {L}_{\mathfrak {s}}=N.\mathcal {L}$. From Lemma 2.8 it follows that singular locus of N coincides with the singular locus of $\mathcal {L}$. Therefore if w is a regular point of $\mathcal {L}$, it is also a regular point of N, hence of $\mathcal {L}_{\mathfrak {s}}$. If else w is a regular singular point of $\mathcal {L}$, then w is either a regular point of $\mathcal {L}_{\mathfrak {s}}$ or a regular singular point of $\mathcal {L}_{\mathfrak {s}}$; in the latter case, by virtue of Proposition 3.9, w is a Fuchsian not-removable singularity. $\quad \square$

Another consequence is an algebraic proof of a well-known Theorem due to L. Fuchs (see for instance [2, Section 5]):

Corollary 3.11

(Fuchs). Consider the scalar differential equation

$$\begin{aligned} y^{(n)}(z)+a_2(z)y^{(n-2)}+\cdots +a_{n}(z)y(z)=0. \end{aligned}$$

The singular point w is a regular singular point for the scalar equation if and only if for every $k=2,\ldots ,n$ the function $(z-w)^{k}a_k(z)$ is analytic at $z=w$.

Proof

Let $\mathfrak {g}=A_{n-1}$, let $V=\mathbb {C}^n$ by the standard representation, and choose as transversal space $\mathfrak {s}$ the space of companion matrices. More precisely, $\mathfrak {s}$ is the space of traceless matrices whose coefficients are all zero outside the first row. We can then choose a basis $\lbrace s_1,\ldots ,s_{n-1} \rbrace$ of $\mathfrak {s}$ such that the scalar equation can be written in the matrix form $\mathcal {L}y=0$, where $\mathcal {L}=\partial _z+f+\sum _k (-1)^ka_{k+1}(z) s_k$. Suppose that w is a regular singular point, namely all solutions have algebraic growth. Then by Lemma 3.7(4), w is a regular singular point of the operator $\mathcal {L}$, and due to Proposition 3.9 it follows that w is a Fuchsian singularity if and only if $(z-w)^ka_k(z)=O(1)$, $\forall k$. Suppose now that $(z-w)^ka_k(z)=O(1)$, $\forall k$. Then by Proposition 3.9w is a Fuchsian singularity of $\mathcal {L}$ hence by Lemma 3.7(1) w is a regular singularity. $\quad \square$

5 Quantum $\widehat{\mathfrak {g}}$-KdV Opers

In this rest of the paper, we develop the following program

• 1. Following [20, 26], we introduce a class of $\mathfrak {g}$-opers,⁴ for $\mathfrak {g}$ simply laced, as the largest class of opers which can provide solutions to the Bethe Ansatz equations. We call them Quantum $\widehat{\mathfrak {g}}$-KdV Opers.

• 2. We prove that these opers actually provide solutions of the Bethe Ansatz equations.

• 3. We characterise these opers explicitly by means of the solution of a fully determined system of algebraic equations.

We recall that in the $\mathfrak {g}=\mathfrak {sl}_2$ case, the above program was addressed and solved in [8] by Bazhanov, Lukyanov, and Zamolodchikov. In this Section, following the proposal of Feigin and Frenkel [20, Section 5] (see also [26, Section 8]), we introduce the Quantum $\widehat{\mathfrak {g}}$-KdV opers in the case $\mathfrak {g}$ is simply laced,⁵ and we give to these opers a first characterisation, which will be used to fully comply with the above program.

5.1 The ground state oper

The Quantum KdV opers are a suitable modification of the simplest opers proposed [20, Section 5], which we studied in our previous papers [42, 43] in collaboration with Daniele Valeri. These opers are expected to correspond to the ground state of the model. Explicitly, they have the form

$$\begin{aligned} \mathcal {L}(x,E)=\partial _x+f+\frac{\ell }{x}+(x^{Mh^\vee }-E) e_\theta , \end{aligned}$$

(4.1)

for arbitrary $\ell \in \mathfrak {h}$ and $M>0$. As observed in [26], after the change of variable

$$\begin{aligned} z=\varphi (x)= \left( \frac{1-\hat{k}}{h^\vee }\right) ^{h^\vee } x^{\frac{h^\vee }{1-\hat{k}}}, \end{aligned}$$

(4.2)

the operator (4.1) is Gauge equivalent to

$$\begin{aligned} \mathcal {L}_G(z,\lambda )&=\partial _z+f+\frac{r}{z}+z^{1-h^\vee }\big ( 1 +\lambda z^{-\hat{k}}\big )e_{\theta }, \end{aligned}$$

(4.3)

which is a form more convenient for the present work. In the above formula $0<\hat{k}<1$, $\lambda \in \mathbb {C}$ and $r \in \mathfrak {h}$ are defined by the relations

$$\begin{aligned} \ell =\frac{h^\vee }{1-\hat{k}} (r-\rho ^\vee )+\rho ^\vee , \qquad M=\frac{\hat{k}}{1-\hat{k}}, \qquad E=-\left( \frac{h}{1-\hat{k}}\right) ^{h\hat{k}}\lambda . \end{aligned}$$

(4.4)

In order to avoid any ambiguity in the definition of the quantum $\widehat{\mathfrak {g}}$-KdV opers, we fix a transversal space $\mathfrak {s}$ and consider the canonical form of the ground-state oper.

Proposition 4.1

The canonical form $\mathcal {L}_{G,\mathfrak {s}}$ of the ground state oper (4.3) is

$$\begin{aligned} \mathcal {L}_{G,\mathfrak {s}}(z,\lambda )=\partial _z+f+\sum _{i=1}^n\frac{\bar{r}^{d_i}}{z^{d_i+1}}+z^{-h^{\vee }+1}(1+ \lambda z^{-\hat{k}})e_{\theta }, \end{aligned}$$

(4.5)

where $\bar{r}=\sum _i\bar{r}^{d_i}$, with $\bar{r}^{d_i}\in \mathfrak {s}^{d_i}$, is the unique element in $\mathfrak {s}$ such that the Lie algebra elements $f-\rho ^\vee +r$ and $f-\rho ^\vee +\bar{r}$ are conjugated.

Proof

The term $z^{1-h^\vee }\big ( 1 +\lambda z^{-\hat{k}}\big )e_{\theta }$ is invariant under unipotent Gauge transformation. Hence, if $\bar{\mathcal {L}}$ is the canonical form of $\partial _z+f+\frac{r}{z}$ then $\mathcal {L}_{G,\mathfrak {s}}=\bar{\mathcal {L}}+z^{1-h^\vee } \big ( 1 +\lambda z^{-\hat{k}}\big )e_{\theta }$. The operator $\partial _z+f+\frac{r}{z}$ is regular in $\mathbb {C}^*$ and has (at most) Fuchsian singularities at $z=0,\infty$. Due to Proposition 3.9, this implies that its canonical form is regular in $\mathbb {C}^*$ and has (at most) Fuchsian singularities at $0,\infty$. Hence it will take the form $\partial _z+f+\sum _{i=1}^n\frac{\bar{r}^{d_i}}{z^{d_i+1}}$ for some $\bar{r}^{d_i} \in \mathfrak {s}^{d_i}$. From Lemma 3.7(3), the principal coefficients at 0 of an operator and of its canonical form are conjugated. Since the principal coefficient at 0 of (4.3) is $f-\rho ^\vee +r$ and that of (4.5) is $f-\rho ^\vee +\bar{r}$, we deduce the thesis. $\quad \square$

We notice here, as it will be important in the next Section, that in any finite dimensional representation the element $f-\rho ^\vee +\bar{r}$ has the same spectrum as $f-\rho ^\vee +r$ (since the two elements are conjugated), which in turn has the same spectrum as $-\rho ^\vee +r$. The latter claim can be proved as follows: in the basis of weight vectors, ordered in decreasing principal order, $-\rho ^\vee +r\in \mathfrak {h}$ is represented by a diagonal matrix and $f\in \mathfrak {n}_-$ is represented by a strictly lower triangular matrix.

Remark 4.2

The operator (4.3) is not meromorphic on the Riemann sphere, because the term $\lambda z^{1-h^\vee -\hat{k}}e_{\theta }$ is multi-valued. However, the element $\lambda z^{1-h^\vee -\hat{k}}e_{\theta }$ is fixed by the action of the Gauge group $\mathcal {N}(K_{\mathbb {P}^1})$, so that it perfectly makes sense to study which properties of $\mathcal {L}_G(z,\lambda )$ are preserved under the action of the meromorphic Gauge groups. Schematically, we have:

$$\begin{aligned} {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\mathbb {P}^1})/\mathcal {N}(K_{\mathbb {P}^1})+\lambda z^{-\hat{k}}e_{\theta } \cong \lbrace {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\mathbb {P}^1})+\lambda z^{-\hat{k}}e_{\theta } \rbrace /\mathcal {N}(K_{\mathbb {P}^1}). \end{aligned}$$

The above comment is consistent with the following fact [26]. Recall the affine Lie algebra $\hat{\mathfrak {g}}$ introduced in Sect. 1, with $d=\lambda \partial _\lambda \in \hat{\mathfrak {g}}$ the corresponding derivation, and set $k=1-h^\vee -\hat{k}$. Then the oper $\mathcal {L}_G(z,\lambda )$ is Gauge equivalent, by means of the affine Gauge transformation $z^{k d}$, to the affine (i.e. $\hat{\mathfrak {g}}$-valued) meromorphic oper

$$\begin{aligned} \partial _z+\hat{f}+\frac{r+k d}{z}+ z^{-h^{\vee }+1 }e_{\theta }, \end{aligned}$$

(4.6)

where $\hat{f}=\sum _{i\in \hat{I}}\hat{f}_i$ is the sum of the negative Chevalley generators of $\hat{\mathfrak {g}}$. In the language of [36], the term $\frac{k}{z}$ is the twist function of the quasi-canonical normal form.

The construction of Bethe Ansatz solutions from $\mathcal {L}_G(z,\lambda )$ can be briefly summarised as follows. The oper $\mathcal {L}_G(z,\lambda )$ has two singular points, $z=0$ and $z=\infty$. The point $z=0$ is a Fuchsian singularity with principal coefficient $f-\rho ^\vee +r$. The point $z=\infty$ is an irregular singularity, with slope $-1+\frac{1}{h^{\vee }}$ and principal term $f+e_{\theta }$. The connection problem between the two singular points is encoded in the Q functions, which we prove to be solutions of the Bethe Ansatz equation for the quantum$-\mathfrak {g}$ KdV model:

$$\begin{aligned} \prod _{j \in I} e^{-2 i \pi \beta _jC_{\ell j}} \frac{Q^{(j)}\Big (e^{i \pi \hat{k}C_{\ell j}}\lambda ^*\Big )}{Q^{(j)} \Big (e^{- i\pi \hat{k}C_{\ell j}}\lambda ^*\Big )}=-1,\qquad i\in I \end{aligned}$$

(4.7)

where $\lambda ^*$ is a zero of $Q^{(i)}(\lambda )$. As it was recalled in the Introduction, the quantum $\widehat{\mathfrak {g}}$-KdV model is specified by a choice of the vacuum parameter $p \in \mathfrak {h}$ and by the central charge c. These determine uniquely the phases $\beta _j$’s of the Bethe Ansatz equations, and the growth order of the solutions $Q^{(i)}$’s for large $\lambda$, see [12]. At the level of the oper (4.5) the phases $\beta _j$’s turn out to be linear functions of the element $r \in \mathfrak {h}$, and the growth order is $\frac{1}{\hat{k}h^{\vee }}$ [12, 43]. Hence the residue at 0 and the slope at $\infty$ fixes uniquely the quantum model.

The natural question is: can the oper $\mathcal {L}_G(z,\lambda )$ be modified in such a way that it still provides solutions of the same Bethe Ansatz equations, possibly corresponding to higher states of the same quantum model? The answer is yes, as we show in the sequel of the paper.

5.2 Higher states: first considerations

Without losing generality, the most general meromorphic deformation of the ground state oper can be written as

$$\begin{aligned} \mathcal {L}(z,\lambda )=\mathcal {L}_{G,\mathfrak {s}}(z,\lambda )+s(z), \end{aligned}$$

(4.8)

where s(z) is an, a priori, arbitrary element of $\mathfrak {s}(K_{\mathbb {P}^1})$. We make four assumptions, equivalent to the ones given in [20, Section 5] (see also [26, Section 8.5]), on the above opers and we show that when these conditions are met solutions of the Bethe Ansatz can be obtained. We thus say that a Quantum $\widehat{\mathfrak {g}}$-KdV oper is an oper of the form (4.8), which satisfies the following assumptions:

Assumption 1

The local structure of the solutions at 0 does not depend on s(z).

Assumption 2

The local structure of the solutions at $\infty$ does not depend on s(z).

Assumption 3

All additional singular points are regular and the corresponding principal coefficients are conjugated to the element $f-\rho ^\vee -\theta ^\vee$.

Assumption 4

All additional singular points have trivial monodromy for every $\lambda \in \mathbb {C}$.

Remark 4.3

These assumptions deserve a brief explanation. The solutions of the Bethe Ansatz equations (4.7) are obtained from $\mathcal {L}_G$ by considering the connection problem between an irregular singularity at $\infty$ and a regular singularity at 0. Moreover, as recalled above, the phases $\beta _j$’s and the order of growth of their solutions $Q^{(i)}$’s are fixed uniquely by the residue at 0 and by the slope at $\infty$. It follows from this that Assumptions 1 and 2 are necessary conditions to obtain solutions of the same Bethe Ansatz equations by the methods developed in [42].

Concerning Assumption 4, if $s \ne 0$, then $\mathcal {L}(z,\lambda )$ has additional singularities, and the connection problem from 0 to $\infty$ is only well defined if these additional singularities have trivial monodromy. In fact, in case of non-trivial monodromy, the connection problem depends on which path in the punctured $\mathbb {C}^*$ one chooses to connect 0 to $\infty$.

We finally discuss Assumption 3. If we assume that the additional singularity is regular, then the triviality of the monodromy (in any representation) implies that the principal coefficient must be conjugated to $f-\rho ^\vee +h$, where h belongs to the co-root lattice of $\mathfrak {g}$, see the discussion before Proposition 8.3. According to [20], the choice $h=-\theta ^\vee$ is, for generic $r,\hat{k}$, a necessary condition for having trivial monodromy for any $\lambda \in \mathbb {C}$. In the case $\mathfrak {g}=\mathfrak {sl}_2$, the latter statement can be verified using elementary considerations, see [8, Appendix B], which is in turn based on [17]. It would be very interesting to clarify Assumption 3 in the general case by means of a similar derivation; we will consider this problem in a forthcoming publication.

We remark that, for the sake of the ODE/IM correspondence, the existence of non-generic opers (i.e. with $h\ne -\theta ^\vee$) may be actually immaterial. In fact, if the ODE/IM correspondence holds true, such non-generic opers, and Bethe Ansatz solutions attached to them, will not presumably correspond to a state of the generalised quantum KdV model. The same remark is valid also for the case of an additional monodromy-free irregular singularity; moreover, we are not aware of any result in the literature about this case and we will not pursue this possibility here.

We organize our analysis of Quantum KdV opers as follows. In the remaining part of the present section we classify the canonical form of opers of type (4.8) satisfying Assumption 1, 2 and 3. In Sect. 5 we construct solutions of the Bethe Ansatz equations when the fourth postulate is met. In Sect. 6 we prove that the canonical form of the quantum KdV opers is Gauge equivalent to a form where all regular singularities are first order pole. The remaining sections of the paper are devoted to the analysis of Assumption 4, and thus to the complete classification of the Quantum–KdV opers.

5.3 The first three assumptions

We provide a more rigorous description of Assumption 1 and 2 by means of the following definition.

Definition 4.4

Let $\mathcal {L}$ be given by (4.8), for some $s\in \mathfrak {s}(K_{\mathbb {P}^1})$. We say that s is subdominant with respect to $\mathcal {L}_{G,\mathfrak {s}}$ at 0 (resp. at $\infty$) if the slope and the principal coefficient of the singularities at 0 (resp. at $\infty$) of $\mathcal {L}$ does not depend on s.

Lemma 4.5

Let $s\in \mathfrak {s}(K_{\mathbb {P}^1})$, and write it as $s=\sum _{i=1}^ns^{d_i}$, with $s^{d_i}\in \mathfrak {s}^{d_i}(K_{\mathbb {P}^1})$. Then s is subdominant with respect to $\mathcal {L}_{G,\mathfrak {s}}$ at 0 if and only if

$$\begin{aligned} s^{d_i}(z)=O(z^{-d_i}),\qquad z\rightarrow 0, \end{aligned}$$

(4.9)

and it is subdominant with respect to $\mathcal {L}_{G,\mathfrak {s}}$ at $\infty$ if and only if

$$\begin{aligned} s^{d_i}(z)=O(z^{-d_i-1}),\qquad z\rightarrow \infty . \end{aligned}$$

(4.10)

Proof

The slope at 0 of $\mathcal {L}_{G,\mathfrak {s}}$ is 1, thus s is subdominant at 0 if and only if $\lim _{z \rightarrow 0}z^{d_i+1} s^{d_i}(z)=0$ for all $i=1,\ldots ,n$. The slope at $\infty$ of $\mathcal {L}_G$ is $-1+\frac{1}{h^{\vee }}$. Let $s^{d_i}(z)=O(z^{-c_i})$, as $z \rightarrow \infty$. Then (cf. Definition 3.4), s(z) is subdominant at $\infty$ if and only if $\frac{c_i}{d_i+1}>1-\frac{1}{h^{\vee }}$, $\forall i$. In other words $c_i>i+1-\frac{d_i+1}{h^{\vee }}$. Since $c_i \in \mathbb {N}$ and $0\le d_i\le h^{\vee }-1$, the latter inequality is satisfied if and only if $c_i\ge d_i+1$. $\quad \square$

The rational functions $s^{d_i}(z)$ satisfying the conditions of the above lemma can be written using a partial fraction decomposition.

Lemma 4.6

Let $i \in \mathbb {N}$ and f be a rational function such that

• (i) $z^i f(z)$ is regular at $z=0$

• (ii) $z^{i+1}f(z)$ is regular at $z=\infty$

• (iii) f is regular in $\mathbb {C}{\setminus }\lbrace 0,\infty \rbrace$ except for a finite (possibly empty) set of points $\{w_j, j \in J\}$, where f has a pole of order $m(j)\ge 1$.

Then there exist $x_l(j)\in \mathbb {C}$, with $j\in J$ and $0\le l \le m(j)-1$, such that

$$\begin{aligned} f(z) = z^{-i}\sum _{j \in J}\sum _{l=0}^{m(j)-1}\frac{x_l(j)}{(z-w_j)^{m(j)-l}}. \end{aligned}$$

Proof

Let $g(z)=z^i f(z)$. By hypotheses (i), (ii) g has poles only at $w_j, j \in I$. Since $g(\infty )=0$, we can represent g as a simple partial fraction without polynomial terms: there exist $x_l(j)\in \mathbb {C}$, with $j \in J$ and $0\le l \le m(j)-1$, such that $g(z)=\sum _{j \in J}\sum _{l=0}^{m(j)-1}\frac{x_l(j)}{(z-w_j)^{m(j)-l}}$. $\quad \square$

As a corollary, we can write explicitly the canonical form of an operator satisfying Assumptions 1, 2 and 3.

Proposition 4.7

An operator $\mathcal {L}(z,\lambda )$ of the form (4.8) satisfies Assumptions 1, 2, 3 if and only if there exists a (possibly empty) arbitrary finite collection of non-zero mutually distinct complex numbers $\lbrace w_j \rbrace _{j \in J}\subset \mathbb {C}^*$ and a collection $s_l^{d_i}(j)$ of arbitrary elements of $\mathfrak {s}^{d_i}$, with $0 \le l \le d_i$ and $j \in J$, such that

$$\begin{aligned} \mathcal {L}(z,\lambda )&= \partial _z+f+\sum _{i=1}^n\frac{\bar{r}^{d_i}}{z^{d_i+1}}+z^{-h^{\vee }+1} (1+ \lambda z^{-\hat{k}})e_{\theta } + \nonumber \\&\quad +\sum _{j \in J}\sum _{i=1}^{n} z^{-d_i}\sum _{l=0}^{d_i}\frac{s^{d_i}_l(j)}{(z-w_j)^{d_i+1-l}}, \end{aligned}$$

(4.11)

where

• $\bar{r}=\sum _i\bar{r}^{d_i}$ is the unique element in $\mathfrak {s}$ such that the Lie algebra elements $f-\rho ^\vee +r$ and $f-\rho ^\vee +\bar{r}$ are conjugated.

• The element $\bar{s}=\sum _i \frac{s_0^{d_i}(j)}{w_j^{d_i}}$ is independent of $j\in J$, and it is the unique element in $\mathfrak {s}$ such that $f-\rho ^\vee -\theta ^\vee$ and $f-\rho ^\vee +\bar{s}$ are conjugated.

Proof

Part of formula (4.11) was already obtained in Proposition 4.1, when considering the canonical form of the ground state oper $\mathcal {L}_G(z,\lambda )$. Due to Lemma 4.6, Assumptions 1, 2 are satisfied if and only if the function $s^{d_i}(z)$ is of the form

$$\begin{aligned} s^{d_i}(z)=z^{-d_i}\sum _{j \in J}\sum _{l=0}^{m_i(j)-1}\frac{s^{d_i}_l(j)}{(z-w_j)^{m_i(j)-l}}, \end{aligned}$$

(4.12)

for some $m(j)\in \mathbb {N}$, and $s^{d_i}_l(j)\in \mathfrak {s}^{d_i}$. For $j\in J$, the principal coefficient of $w_j$ is given by $f-\rho ^\vee +\bar{s}$, where $\bar{s}=\sum _i \frac{s_0^{d_i}(j)}{w_j^{d_i}}$. Assumption 3 states that for every $j\in J$ the additional singularity $w_j$ has to be regular, and its principal coefficient $f-\rho ^\vee +\bar{s}$ is conjugated to the element $f-\rho ^\vee -\theta ^\vee$. In particular, $\bar{s}$ in independent of $j\in J$. Due to Proposition 3.9, a singular point w for an oper in canonical form is regular if and only if it is Fuchsian, from which it follows that in (4.12) we have $m_i(j)=d_i+1$ for every i, j, proving the proposition. $\quad \square$

6 Constructing Solutions to the Bethe Ansatz

In this section, adapting the techniques of [42], we construct solutions of the Bethe Ansatz equations as coefficients of the central connection problem for opers $\mathcal {L}$ of type (4.8) and satisfying Assumption 1, 2, 3 and 4. According to Proposition 4.7, we restrict our analysis to the subset of operators of the form (4.11) such that all additional singularities $\lbrace w_j \rbrace _{j \in J}$ have trivial monodromy. The latter condition implies that all solutions $\psi$ of the differential equation $\mathcal {L}\psi =0$ are meromorphic functions on the universal cover of $\mathbb {C}^*$, whose (possible) singularities are pole singularities located at the lift of the points $w_j, j \in J$.

Definition 5.1

Let $\widehat{\mathbb {C}}$ be the universal cover of $\mathbb {C}^*$, minus the lift of the points $w_j,j \in J$. If V is a $\mathfrak {g}$-module, and fixed $\lambda \in \mathbb {C}$, we consider solutions of $\mathcal {L}(z,\lambda )\psi (z,\lambda )=0$ as analytic functions $\psi (\cdot ,\lambda ):\widehat{\mathbb {C}}\rightarrow V$.

Remark 5.2

For sake of notation simplicity, we assume a branch cut on the negative real semi-axis and use the coordinate z of the base space for the first sheet of the covering. Whenever we write $f(e^{2\pi i}z)$ we mean that we evaluate the function f on the second sheet. This corresponds to the counter-clockwise analytic continuation of the function f(z) along a simple Jordan curve encircling $z=0$.

Definition 5.3

Let $O_\lambda$ denote the ring of entire functions of the variable $\lambda$. If V is a $\mathfrak {g}$-module, we denote by $V(\lambda )$ the set of solutions of $\mathcal {L}(z,\lambda )\psi (z,\lambda )=0$ which are entire functions of $\lambda$, i.e. they are analytic functions $\psi :\widehat{\mathbb {C}} \times \mathbb {C}\rightarrow V$.

Lemma 5.4

$V(\lambda )$ is a free module over the ring $O_\lambda$, and its rank is the dimension of V. That is $V(\lambda )\cong V \otimes _{\mathbb {C}}O_\lambda$.

Proof

In order to find an $O_\lambda$-basis of solutions it is sufficient to find a set $\lbrace \psi _i(z,\lambda ),i=1,\ldots ,\dim V\rbrace$ of elements in $V(\lambda )$ which is a $\mathbb {C}$-basis of solutions of $\mathcal {L}(z,\lambda )\psi (z,\lambda )=0$ for every fixed $\lambda$. Let then $\lbrace \psi _i,i=1,\ldots ,\dim V\rbrace$ be a basis of V. Fix a regular point $z_0$ and let $\psi _i(z,\lambda )$ be the solution of $\mathcal {L}\psi =0$ satisfying the Cauchy problem $\psi _i(z_0,\lambda )=\psi _i$ for all $\lambda \in \mathbb {C}$. The solutions $\psi _i(z,\lambda ) \in V(\lambda )$ because the differential equation depends analytically on $\lambda$, and are—by construction—a basis of V for each fixed $\lambda$. $\quad \square$

For $t\in \mathbb {R}$, we define the following twisted operator and twisted solution:

$$\begin{aligned}&\mathcal {L}^t(z,\lambda )=\mathcal {L}(e^{2i \pi t}z,e^{2i \pi t \hat{k}}\lambda ) \end{aligned}$$

(5.1)

$$\begin{aligned}&\psi _{t}(z,\lambda )=e^{2 i \pi t \rho ^\vee }\psi (e^{2\pi i t}z, e^{2 \pi i t \hat{k}}\lambda ) \end{aligned}$$

(5.2)

Applying the change of variable formula (2.8) to (4.11), we have that

$$\begin{aligned} \mathcal {L}^t(z,\lambda )=\partial _z+f+\sum _i\frac{\bar{r}^{d_i}}{z^{d_i+1}}+e^{- 2\pi i t } z^{1-h^{\vee }}\big (1+\lambda z^{-\hat{k}} \big )e_{\theta }+ \sum _{m=1}^n e^{2\pi i t (d_m+1)} s^{d_m}(e^{2\pi i t}z) \;, \end{aligned}$$

(5.3)

where

$$\begin{aligned} s^{d_m}(z)= z^{-d_m}\sum _{j \in J}\sum _{l=0}^{d_m}\frac{s^{d_m}_l(j)}{(z-w_j)^{d_m+1-l}}. \end{aligned}$$

(5.4)

By the same formula, it is straightforward to see that the twisted function $\psi _t(z,\lambda )$ is a solution of the twisted operator: $\mathcal {L}^t(z,\lambda )\psi _t(z,\lambda )=0$. Note that when $t=1$ then $\mathcal {L}^{1}(z,\lambda )=\mathcal {L}(z,\lambda )$, while in general $\psi _1(z,\lambda )$ is not equal to $\psi (z,\lambda )$. We define on $V(\lambda )$ the following $O_\lambda$-linear operator, the monodromy operator:

$$\begin{aligned} M: V(\lambda ) \rightarrow V(\lambda ), \qquad M(\psi )(z,\lambda )= e^{2i\pi \rho ^\vee }\psi (e^{2\pi i}z,e^{2\pi i \hat{k} }\lambda ). \end{aligned}$$

(5.5)

Remark 5.5

Since $\mathcal {L}(z,\lambda )$ is a multivalued function of z, the monodromy operator cannot be defined on solutions $\mathcal {L}(z,\lambda )\psi (z)=0$ for fixed $\lambda$.

Remark 5.6

In the case of the ground state, we have

$$\begin{aligned} \mathcal {L}_{G,\mathfrak {s}}^t(z,\lambda )=\partial _z+f+\sum _i\frac{\bar{r}^{d_i}}{z^{d_i+1}}+ e^{- 2\pi i t } z^{1-h^{\vee }}\big (1+\lambda z^{-\hat{k}} \big )e_\theta . \end{aligned}$$

Hence the the twist by t is tantamount to a change $e_{\theta }\rightarrow e^{- 2\pi i t } e_{\theta }$, which in turn can be interpreted as an automorphism of a Kac Moody algebra with a different loop variable than $\lambda$. This is the point of view that we used in our previous paper. We drop this interpretation in the present work, for it cannot be simply extended to the more general operators we are considering.

6.1 The n fundamental modules

Let $\omega _i$ be the i-th fundamental weight of the Lie algebra $\mathfrak {g}$, and denote by $L(\omega _i)$ the i-th fundamental representation of $\mathfrak {g}$. Let the function $p: I \rightarrow \mathbb {Z}/{2\mathbb {Z}}$ be defined inductively as follows: $p(1)=0$, and $p(j)=p(i)+1$ whenever $C_{ij}<0$ (i.e. p alternates on the Dynkin diagram). Note that the principal coefficient at $\infty$ of (4.11) is given by $f+e_{\theta }$, where $f=\sum _{i\in I}f_i$ and $f_i$, $i\in I$, are fixed (negative) Chevalley generators of $\mathfrak {g}$. The highest root vector $e_{\theta }$ is defined up to a scalar multiple, and the spectrum of the principal coefficient $f+e_\theta$ in $L(\omega _i)$—hence the asymptotic behaviour of solutions $\mathcal {L}\psi =0$—depends on such a choice. We choose it according to the following Proposition.

Proposition 5.7

[42, Proposition 4.4]. One can choose the element $e_{\theta }$ in such a way that for every $i \in I$ the linear operator representing $f+(-1)^{p(i)}e_{\theta }$ in $L(\omega _i)$ has a unique eigenvalue $\lambda ^{(i)}$ with maximal real part, which is furthermore real, positive, and simple. In fact, the array of eigenvalues $(\lambda ^{(1)},\ldots ,\lambda ^{(n)})$ can be characterised as the Perron–Frobenius eigenvector of the incidence matrix $B=2 \mathbb {1}_n-C$ of the Dynkin diagram of $\mathfrak {g}$:

$$\begin{aligned} \sum _j B_{ij} \lambda ^{(j)}=2 \cos (\frac{\pi }{h^{\vee }}) \lambda ^{(i)},\quad \lambda ^{(1)}=1, \quad B_{ij}=2 \delta _{i,j}-C_{ij}. \end{aligned}$$

(5.6)

Note that due to the definition of p(i), from (5.3) it follows that for $i\in I$ we have

$$\begin{aligned} \mathcal {L}^{\frac{p(i)}{2}}(z,\lambda )&=\partial _z+f+\sum _{m=1}^n\frac{\bar{r}^{d_m}}{z^{d_m+1}}+(-1)^{ p(i) } z^{1-h^{\vee }}\big (1+\lambda z^{-\hat{k}} \big )e_{\theta }\nonumber \\&\quad +\sum _{m=1}^nz^{-d_m}\sum _{j \in J}\sum _{l=0}^{d_m}\frac{s^{d_m}_l(j)}{(z+(-1)^{p(i)+1}w_j)^{d_m+1-l}}. \end{aligned}$$

(5.7)

Definition 5.8

Fixed $e_{\theta }$ as in Proposition 5.7 above, for each $i\in I$ we set $V^i(\lambda )$ to be the $O_\lambda$-module of solutions of the differential equation

$$\begin{aligned} \mathcal {L}^{\frac{p(i)}{2}}(z,\lambda )\psi (z,\lambda )=0, \qquad \psi (z,\lambda ): \widehat{\mathbb {C}}\times \mathbb {C} \rightarrow L(\omega _i), \end{aligned}$$

(5.8)

where $\mathcal {L}^{\frac{p(i)}{2}}$ is given by (5.7).

6.2 The singularity at 0

In order to study the monodromy of solutions about $z=0$, we address the local behaviour of solution in a neighbourhood of the singular point $z=0$. Applying $z^{{{\,\mathrm{ad}\,}}\rho ^\vee }$ to (5.7) (see equation (3.3)), we have that

$$\begin{aligned} z^{{{\,\mathrm{ad}\,}}\rho ^{\vee }} \mathcal {L}^{\frac{p(i)}{2}}= \partial _z+ \frac{f-\rho ^{\vee } + \bar{r}}{z} +(-1)^{p(i)}e_{\theta }\lambda z^{-\hat{k}}+O(1), \end{aligned}$$

(5.9)

where $\bar{r}=\sum _i \bar{r}^{d_i}$. Hence, $z=0$ is a Fuchsian singular point (cf. Definition 3.5), and the principal coefficient $f-\rho ^{\vee } + \bar{r}$ is independent on the sign p(i).

We remark again that the singularity at 0 is also a ramification point of the potential. If $\hat{k}$ is rational, then the operator $\mathcal {L}$ can be made single-valued by a change of variable, and the standard Frobenius method applies. This is the case we considered in [42]. If $\hat{k}$ is irrational then the operator cannot be made single-valued by a change of variables. Therefore the standard theorems on Frobenius series do not apply. We develop below an appropriate modification of the Frobenius method in the case $\hat{k}$ is irrational and satisfies a genericity assumption that implies that no logarithms are present in the local expansion at 0. Filling this gap, we also complete our previous works.

The local behaviour of the solutions at 0 depends on the spectrum of $f-\rho ^{\vee } +\bar{r}$ in the representation we are considering. As we remarked earlier this element is conjugated to $f-\rho ^\vee +r$, which has the same spectrum as $r-\rho ^\vee \in \mathfrak {h}$. Recall the weight lattice P introduced in (1.2). If $P_{\omega _i}\subset P$ denotes the set of weights of the representation $L(\omega _i)$, then the spectrum of $f-\rho ^{\vee } + \bar{r}$ in this representation is the set $\lbrace \omega (r-\rho ^\vee ), \omega \in P_{\omega _i} \rbrace$. In order to proceed, we need to consider separately the case when $\hat{k}\in (0,1)$ is rational and the case when it is irrational. First, we have

Definition 5.9

Let $i\in I$, and let $P_{\omega _i}\subset P$ be the weights of the fundamental representation $L(\omega _i)$. Moreover, put $T=\{a=n+m\hat{k}\,:\, n,m\in \mathbb {Z}, (n,m)\ne (0,0)\}\subset \mathbb {C}$. If $\hat{k}$ is irrational then the pair $(r,\hat{k})\in \mathfrak {h}\times (0,1)$ is said to be generic if, for every $i \in I$ and for every $\omega \in P_{\omega _i}$, the spectrum of the matrix $r-\rho ^{\vee }-\omega (r-\rho ^{\vee })\text {Id}$ in the representation $L(\omega _i)$ is contained in $\mathbb {C}{\setminus } T.$ If $\hat{k}$ is rational, say $\hat{k}=p/q$ with $q>p \in \mathbb {N}$, then we say that the pair $(r,\hat{k})\in \mathfrak {h}\times (0,1)$ is generic if, for any $i\in I$ and for every $\omega \in P_{\omega _i}$, the spectrum of the matrix $q (r-\rho ^\vee )-q\omega (r-\rho ^\vee ) \text {Id}$ in the representation $L(\omega _i)$ is contained in $\mathbb {C}{\setminus } \mathbb {Z}_{>0}.$

We have the following result.

Proposition 5.10

Let $(r,\hat{k})\in \mathfrak {h} \times (0,1)$ be a generic pair. Let $\big (\omega (r-\rho ^\vee ),\chi _{\omega }\big )$ be an eigenpair composed of an eigenvalue and a corresponding eigenvector of $f-\rho ^\vee +\bar{r}$ in $L(\omega _i)$.

A unique solution $\chi _{\omega }(z,\lambda )$ in $V^i(\lambda )$ is determined by the following expansion at (0, 0):

$$\begin{aligned} \chi _{\omega }(z,\lambda )=z^{- \rho ^\vee }z^{-\omega (r-\rho ^\vee )} F(z,\lambda z^{-\hat{k}} ), \end{aligned}$$

(5.10)

where $F(z,\eta )$ is an $L(\omega _i)$-valued function, analytic in a neighbourhood of (0, 0) such that $\lim _{z \rightarrow 0} F(z,\lambda z^{-\hat{k}})=\chi _{\omega }.$

Proof

For the case of $\hat{k}$ irrational the proof is in the Appendix.

The case of a rational $\hat{k}$ was proven in [42, Section 5]. We sketch here the proof, and the reader can consult the cited paper for more details. Applying the transformation (2.7) with $z=\varphi (x)=x^{q}$ to the operator $z^{\rho ^\vee }\mathcal {L}^{\frac{p(i)}{2}}$ (5.9), one obtains

$$\begin{aligned} \widetilde{\mathcal {L}}&=\partial _x+q\,\frac{f- r-\rho ^\vee }{x}+b(x). \end{aligned}$$

where $b \in \mathfrak {b}_+(K_{\mathbb {P}^1})$ is regular at 0.

The standard Frobenius method provide local solutions of the equation $\widetilde{\mathcal {L}} \psi =0$ in a neighbourhood of $x=0$: Provided the difference between any two eigenvalues of the matrix $q (f- r-\rho ^\vee )$ is not a positive number, i.e. provided $(r,\hat{k})$ is generic, to any eigenpair $(q\omega (r-\rho ^\vee ,q\chi _{\omega })$ of $q (f- r-\rho ^\vee )$ there corresponds a convergent Frobenius solution of the form

$$\begin{aligned} \widetilde{\chi _{\omega }}(x,\lambda )=x^{-q \omega (r-\rho ^\vee )}\big ( \chi _{\omega } +\sum _{j \ge 1} a_j x^j\big ) \end{aligned}$$

It follows that the equation $\mathcal {L}^{\frac{p(i)}{2}}\psi =0$ admits the solution

$$\begin{aligned} \chi _{\omega }(z,\lambda )= z^{-\rho ^\vee }\widetilde{\chi _\omega }(z^{\frac{1}{q}},\lambda )=z^{-\rho ^\vee }z^{-\omega (r-\rho ^\vee )} \big ( \chi _{\omega } +\sum _{j \ge 1} a_j z^{\frac{j}{q}}\big ). \end{aligned}$$

Moreover, a closer inspection of the series $\chi _{\omega }+\sum _{j \ge 1} a_j z^{\frac{j}{q}}$ shows that it is of the required form. $\quad \square$

A direct computation shows that the solution $\chi _{\omega }(z,\lambda )$ of Proposition 5.10 is an eigenfunction of the monodromy operator (5.5). It follows from this that if $r+\rho ^\vee -f$ is semi-simple (i.e. diagonalizable) then solutions of the form (5.10) are an eigenbasis of the monodromy operator in the module $V^i(\lambda )$, for $i\in I$. More precisely, we have:

Corollary 5.11

Let $(r,\hat{k})$ be a generic pair, and $\chi _{\omega }(\lambda ,z)$ be the solution constructed in Proposition 5.10. Then

$$\begin{aligned} M(\chi _{\omega })(z,\lambda )= e^{2\pi i\rho ^\vee } \chi _{\omega }\big (e^{2 \pi i}z,e^{2 \pi i\hat{k}}\lambda \big )=e^{2i\pi \omega (\rho ^\vee -r) } \chi _{\omega }\big (z,\lambda \big ), \end{aligned}$$

(5.11)

which means that $\chi _{\omega }(z,\lambda )$ is an eigenfunction of the monodromy operator. Let moreover $f-\rho ^\vee +\bar{r}$ be semisimple, and $\lbrace \chi _{\omega }\rbrace _{\omega \in P_{\omega _i}}$—taking into consideration weight multiplicities—be a basis of $L(\omega _i)$ made of eigenvectors of $f-\rho ^\vee +\bar{r}$. Then $\lbrace \chi _{\omega }(z,\lambda )\rbrace _{\omega \in P_{\omega _i}}$ is a $O_\lambda$-basis of $V^i(\lambda )$.

Proof

The first part of the Lemma is a direct consequence of the Proposition. If $f-\rho ^\vee +\bar{r}$ is semisimple, then it admits a basis of eigenvectors $\chi _{\omega },\omega \in P_{\omega _i}$. It follows that $\lbrace \chi _{\omega }(z,\lambda )\rbrace _{\omega \in P_{\omega _i}}$ is a $\mathbb {C}$ basis of solutions for each fixed $\lambda$, and hence a $O_\lambda$-basis of $V^i(\lambda )$. $\quad \square$

We finally consider the transformation of solutions of type (5.10) under Gauge transformations. These results will be useful later, to show that the Q-functions (which satisfy the Bethe Ansatz) are Gauge invariant. In other words the Q-functions are properties of the opers, and not just of the single differential operators.

Let $y\in \mathfrak {n}_+(K_{\mathbb {P}^1})$, so that $\exp (y)\in \mathcal {N}(K_{\mathbb {P}^1})$, and denote $\bar{\mathcal {L}}^{\frac{p(i)}{2}}=\exp ({{\,\mathrm{ad}\,}}y).\mathcal {L}^{\frac{p(i)}{2}}$, and $\bar{\chi }_{\omega }=\exp (y).\chi _{\omega }$. By construction we have $\bar{\mathcal {L}}^{\frac{p(i)}{2}}\bar{\chi }_\omega =0$, and since y is meromorphic, applying the monodromy operator (5.5) on $\bar{\chi }_{\omega }$ and using (5.11) we get $M(\bar{\chi }_{\omega })=e^{-2i\pi \omega (r-\rho ^\vee ) }\bar{\chi }_{\omega }$. Thus, solutions of type (5.10) satisfy the relation

$$\begin{aligned} M(\exp (y).\chi _\omega )=\exp (y).M(\chi _\omega ), \qquad y \in \mathfrak {n}_+(K_{\mathbb {P}^1}). \end{aligned}$$

In addition, if $\lbrace \chi _{\omega }(z,\lambda )\rbrace _{\omega \in P_{\omega _i}}$ is a $O_\lambda$-basis of solutions for the equation $\mathcal {L}^{\frac{p(i)}{2}}\psi =0$ (namely, a basis for the $O_\lambda$-module $V(\lambda )^i$ introduced in Definition 5.8), then $\lbrace \exp (y).\chi _{\omega }(z,\lambda )\rbrace _{\omega \in P_{\omega _i}}$ is a $O_\lambda$-basis of solutions for the equation $\bar{\mathcal {L}}^{\frac{p(i)}{2}}\psi =0$.

6.3 The singularity at $\infty$

Now we move to the analysis of the irregular singularity at $\infty$. In order to compute the asymptotic behaviour of solutions around $\infty$, we define the function $q(z,\lambda )$ as the truncated Puiseaux series that coincides with $\big (z^{1-h^\vee }(1+\lambda z^{-\hat{k}})\big )^{\frac{1}{h^\vee }}$ up to a remainder $o(z^{-1})$:

$$\begin{aligned} q(z,\lambda )=z^{\frac{1}{h^\vee }-1} \big ( 1 + \sum _{l=1}^{\lfloor \frac{1}{\hat{k}h^\vee } \rfloor } c_l \lambda ^l z^{-l \hat{k}}\big ). \end{aligned}$$

(5.12)

Here $c_l$ are the coefficients of the MacLaurin expansion of $(1-w)^{\frac{1}{h^\vee }}$.⁶ If we apply the Gauge transformation $q(z,\lambda )^{-{{\,\mathrm{ad}\,}}\rho ^{\vee }}$ to the operator (5.7) we obtain

$$\begin{aligned} q(z,\lambda )^{-{{\,\mathrm{ad}\,}}\rho ^{\vee }}.\mathcal {L}^{\frac{p(i)}{2}}(z,\lambda )=\partial _z+ q(z,\lambda ) \Lambda ^i + O(z^{-1-\varepsilon }), \end{aligned}$$

(5.13)

where $\Lambda ^i=f+(-1)^{p(i)}e_{\theta }$ is (the image in the evaluation representation of) the cyclic element [32, §14] of the Kac–Moody algebra $\hat{\mathfrak {g}}$, and $\varepsilon$ a positive real number.

The transformed operator (5.13) has Poincaré rank $\frac{1}{h^{\vee }}$ with semi-simple principal coefficient $\Lambda ^i$, and a perturbation which is integrable at $\infty$. It follows that the dominant part of the asymptotic expansion of the solutions near $\infty$ is fully characterised by the spectrum of the principal coefficient [45]. In particular, the subdominant behaviour as $z \rightarrow + \infty$ is dictated by the eigenvector with maximal real part. Indeed, we have the following proposition which is adapted from [42, Theorem 3.4].

Proposition 5.12

Let $\psi ^{(i)} \in L(\omega _i)$ be an eigenvector of $f+(-1)^{p(i)}e_{\theta }$ with the maximal eigenvalue $\lambda ^{(i)}$, as defined in Proposition 5.7.

• 1. For every $\lambda \in \mathbb {C}$ there exists a unique solution $\Psi ^{(i)}(z,\lambda )$ such that

  $$\begin{aligned} \Psi ^{(i)}(z,\lambda )=e^{-\lambda ^{(i)} S(z,\lambda )}q(z,\lambda )^{\rho ^\vee } \left( \psi ^{(i)} +o(1) \right) , \end{aligned}$$

  (5.14)

  as $z \rightarrow +\infty$, where $S(z,\lambda )=\int _{z_0}^z q(w,\lambda ) dw$ for some $z_0 \in \mathbb {C}$.

• 2. For any $\lambda \in \mathbb {C}$, if $\psi (\cdot ,\lambda ): \mathbb {C} \rightarrow L(\omega _i)$ is a non-trivial solution of $\mathcal {L}^{\frac{p(i)}{2}}\psi =0$ then $\Psi ^{(i)}(z,\lambda )=o\big (\psi (z,\lambda )\big )$ as $z\rightarrow \infty$ unless $\psi (z,\lambda )= C \Psi ^{(i)}(z,\lambda )$ for a $C \in \mathbb {C}^*$.

• 3. $\Psi ^{(i)}(z,\lambda )\in V^i(\lambda )$, i.e. it depends analytically on $\lambda$.

Proof

It follows from (5.13) and Proposition 5.7. The detailed proof can be found in [42, Theorem 3.4]. $\quad \square$

By means of the characterization given in Theorem 5.12(2) above, we can define the subdominant solution $\Psi ^{(i)}$ for any choice operator Gauge equivalent to $\mathcal {L}^{\frac{p(i)}{2}}$. Let $\exp (y)\in \mathcal {N}(K_{\mathbb {P}^1})$, and denote $\bar{\mathcal {L}}^{\frac{p(i)}{2}}=\exp ({{\,\mathrm{ad}\,}}y).\mathcal {L}^{\frac{p(i)}{2}}$. Then there is a unique (up to a scalar mutliple) solution $\overline{\Psi }^{(i)}(z,\lambda )$ of $\bar{\mathcal {L}}^{\frac{p(i)}{2}}\psi =0$, belonging to $V^{i}(\lambda )$, and satisfying Theorem 5.12(2). This is indeed $\exp ( y) \Psi ^{i}(z,\lambda )$.

6.4 The $\Psi$-system

The next, and main algebraic step, towards constructing solutions of the Bethe Ansatz equations is the $\Psi$-system, derived in [42], which the reader should consult for all details.

Let $i\in I$, recall the definition of the incidence matrix $B=2\mathbb {1}_{n}-C$, and consider the $\mathfrak {g}$-modules $\bigwedge ^2 L(\omega _i)$ and $\bigotimes _{j\in I}L(\omega _j)^{\otimes B_{ij}}$. These are, in general, not isomorphic, but they have the same highest weight $\eta _i=\sum _j B_{ij} \omega _j$, which has multiplicity one. Now for every $j\in I$ fix a highest weight vector $\psi _{\omega _j}$ of $L(\omega _j)$, and set $\psi _{\omega _j-\alpha _j}=f_j\psi _{\omega _j}$. Then, for every $i\in I$ we have that $\psi _{\omega _i}\wedge \psi _{\omega _i-\alpha _i}$ is an highest weight vector of $\bigwedge ^2 L(\omega _i)$, of weight $\eta _i$, while $\bigotimes _{j\in I}\psi _{\omega _j}^{\otimes B_{ij}}$ is an highest weight vector of $\bigotimes _{j\in I}L(\omega _j)^{\otimes B_{ij}}$, of weight $\eta _i$. Hence, for every $i\in I$, we have a well-defined homomorphisms of representations

$$\begin{aligned} m_i: \bigwedge ^2 L(\omega _i) \rightarrow \bigotimes _{j\in I}L(\omega _j)^{\otimes B_{ij}}, \quad m_i(\psi _{\omega _i}\wedge \psi _{\omega _i-\alpha _i})= \bigotimes _{j\in I}\psi _{\omega _j}^{\otimes B_{ij}}, \end{aligned}$$

(5.15)

uniquely defined by requiring that it annihilates the (possibly trivial) submodule $U_i \subset \bigwedge ^2 L(\omega _i)$ such that $\bigwedge ^2L(\omega _i)=L(\eta _i)\oplus U_i$.

Proposition 5.13

Let $\Psi ^{(i)}(z,\lambda )$ be the sub-dominant solution defined in Proposition 5.12, and $\Psi ^{(i)}_{\frac{1}{2}}(z,\lambda )$ the same solution, twisted according to formula (5.2). We can choose a normalisation of the solutions $\Psi ^{(i)}(z,\lambda )$’s in such a way that the following set of identities—known as $\Psi$-system—holds true:

$$\begin{aligned} m_i\big ( \Psi _{-\frac{1}{2}}^{(i)}(z,\lambda ) \wedge \Psi _{\frac{1}{2}}^{(i)}(z,\lambda ) \big ) =\otimes _{j\in I} \Psi ^{(j)}(z,\lambda )^{\otimes B_{ij}}\,, \quad i\in I, \end{aligned}$$

(5.16)

where $m_i$ is the morphism of $\mathfrak {g}$ modules defined in (5.15).

Proof

It follows from Proposition 5.12. See [42, Theorem 3.6] for details. $\quad \square$

6.5 The $Q\widetilde{Q}$ system and the Bethe Ansatz

We are now in the position of proving the $Q\widetilde{Q}$ system, which implies the Bethe Ansatz equations. We suppose that $(r,\hat{k})$ is a generic pair and that $f+r-\rho ^\vee$ is semisimple, so that, after Corollary 5.11, the set $\lbrace \chi _{\omega }(z,\lambda )\rbrace _{\omega \in P_{\omega _i}}$ is a $O_\lambda$-basis of $V^i(\lambda )$ (weight-multiplicity is considered). Therefore we have the following decomposition

$$\begin{aligned} \Psi ^{(i)}(z,\lambda )=\sum _{\omega \in P_{\omega _i}}Q_{\omega }(\lambda ) \chi _{\omega }(z,\lambda ), \qquad i\in I, \end{aligned}$$

(5.17)

where the coefficients $Q_{\omega }(\lambda )$ are entire functions of $\lambda$. Note that $\omega _i$ has mutliplicity 1, as does any weight of the form $\sigma (\omega _i)$ for all $\sigma \in \mathcal {W}$, where $\mathcal {W}$ is the Weyl group of $\mathfrak {g}$; in particular the weight $\omega _i-\alpha _i$ belongs to the $\mathcal {W}$ orbit of $\omega _i$.

We show that, as a direct consequence of the $\Psi$-system, the coefficients $Q_{\omega }(\lambda )$ satisfy the $Q\widetilde{Q}$-system and thus the Bethe Ansatz equations.

Since after a Gauge transformation $N=e^{{{\,\mathrm{ad}\,}}y},y \in \mathfrak {n}^+(K_{\mathbb {P}^1})$, the solution $\Psi ^{(i)}$ and the solutions $\chi _{\omega }$ transforms as vectors, namely $\Psi ^{(i)} \rightarrow e^{y}\Psi ^{(i)}$, $\chi _{\omega }\rightarrow e^{y}\chi _{\omega }$, it follows immediately that the entire functions $Q_\omega (\lambda )$, $\omega \in P_{\omega _i}, i \in I$ are invariant under Gauge transformations. This shows that the solutions of the Bethe Ansatz equations we construct are not just a properties of the operator $\mathcal {L}\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\mathbb {P}^1})$, but of the oper $[\mathcal {L}]\in {{\,\mathrm{Op}\,}}_{\mathfrak {g}}(\mathbb {P}^1)$.

Theorem 5.14

Suppose that the pair $(r,\hat{k}) \in \mathfrak {h}\times (0,1)$ is generic, and $f+r-\rho ^\vee$ is semisimple, so that the decomposition (5.17) holds. Fix an arbitrary element $\sigma$ of the Weyl group $\mathcal {W}$ of $\mathfrak {g}$, and for every $\ell \in I$ denote $Q_\sigma ^{(\ell )}=Q_{\sigma (\omega _\ell )}$ and $\widetilde{Q}^{(\ell )}_\sigma =Q_{\sigma (\omega _\ell -\alpha _\ell )}$.

One can normalise the solutions $\chi _{\sigma (\omega _\ell )},\chi _{\sigma (\omega _\ell -\alpha _\ell )},\ell \in I$ so that the following identity—known as $Q\widetilde{Q}$-system—holds for every $\ell \in I$

$$\begin{aligned} \prod _{j \in I}\left( Q^{(j)}_\sigma (\lambda )\right) ^{B_{\ell j}}&=e^{i \pi \theta _\ell } Q^{(\ell )}_\sigma (e^{-\pi i \hat{k}}\lambda )\widetilde{Q}_\sigma ^{(\ell )}(e^{\pi i \hat{k}}\lambda ) \nonumber \\&-e^{-i \pi \theta _\ell } Q_\sigma ^{(\ell )}(e^{\pi i \hat{k}}\lambda )\widetilde{Q}_\sigma ^{(\ell )}(e^{-\pi i \hat{k}}\lambda ) \,, \end{aligned}$$

(5.18)

where $\theta _\ell =\sigma (\alpha _\ell )(r-\rho ^\vee )$.

Proof

It is a straightforward computation: plug the decomposition (5.17) and the expansion (5.10) into the $\Psi$-system (5.16). $\quad \square$

It might be useful to recall that in equation (5.18) we have $B_{ij}\in \{0,1\}$, due to the fact that $\mathfrak {g}$ is simply laced.

Remark 5.15

The $Q\widetilde{Q}$-system, first obtained in [42, 43], was shown in [26] to be a universal system of relations in the commutative Grothendieck ring $K_0(\mathcal {O})$ of the category $\mathcal {O}$ of representations of the Borel subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak {g}})$.

The Bethe Ansatz equation is a straightforward corollary of the $Q\widetilde{Q}$ system.

Corollary 5.16

Let $(r,\hat{k})$ be a generic pair. Let us assume that the functions $Q^{(i)}_\sigma (\lambda )$ and $\widetilde{Q}^{(i)}_\sigma (\lambda )$ do not have common zeros. For any zero $\lambda ^*_i$ of $Q^{(i)}(\lambda )$, the following system of identities—known as $\mathfrak {g}$-Bethe Ansatz—holds

$$\begin{aligned} \prod _{j = 1}^{n} e^{-2 i \pi \beta _jC_{\ell j}} \frac{Q_\sigma ^{(j)}\Big (e^{i \pi \hat{k}C_{\ell j}}\lambda _\ell ^*\Big )}{Q_\sigma ^{(j)} \Big (e^{- i\pi \hat{k}C_{\ell j}}\lambda _\ell ^*\Big )}=-1 \,, \end{aligned}$$

(5.19)

with $\beta _j=\sigma (\omega _j)(r-\rho ^\vee )$.

Remark 5.17

What happens when $(r,\hat{k})$ is a non-generic pair? In that case in general the monodromy operator is not diagonalizable. Hence we can define the coefficients $Q^{(j)}_{\sigma }$ functions only for $\sigma$ belonging to a proper subset of $\mathcal {W}$ [42, 43]. The same phenomenon occurs also at the level of the quantum KdV model: for some values of $(r,\hat{k})$ not all Q functions can be defined. Hence the ODE/IM correspondence is expected to hold also for non-generic values of the parameters.

7 Extended Miura Map for Regular Singularities

Due to Proposition 4.7, a $\mathfrak {g}$-oper $\mathcal {L}(z,\lambda )$ of type (4.8) satisfying Assumptions 1, 2, 3 can be written in the canonical form (4.11). We prove in this section that the $\mathfrak {g}$-oper $\mathcal {L}(z,\lambda )$ admits the representation (0.3). We prove in fact the following:

Theorem 6.1

Fix $(r,\hat{k}) \in \mathfrak {h}\times (0,1)$. Any operator $\mathcal {L}(z,\lambda )$ satisfying Assumptions 1, 2, 3 defining the quantum $\widehat{\mathfrak {g}}$-KdV opers, with a (possibly empty) set $\{w_j\,,\,j\in J\}$ of additional poles, is Gauge equivalent to a unique operator of the form

$$\begin{aligned} \mathcal {L}(z,\lambda )= \partial _z+f+\frac{r}{z}+\sum _{j \in J}\frac{1}{z-w_j}\left( -\theta ^\vee +\sum _{i=1}^{h^\vee -1} \frac{X^i(j)}{z^i} \right) +z^{-h^{\vee }+1} (1+ \lambda z^{-\hat{k}})e_{\theta }, \end{aligned}$$

(6.1)

for some $X^i(j) \in \mathfrak {g}^i$, with $i=1,\ldots ,h^\vee -1$ and $j\in J$.

We begin our analysis by decomposing the normal form (4.8) as

$$\begin{aligned} \mathcal {L}(z,\lambda )=\mathcal {L}_\mathfrak {s}(z)+z^{-h^{\vee }+1}(1+ \lambda z^{-\hat{k}})e_{\theta }, \end{aligned}$$

where

$$\begin{aligned}&\mathcal {L}_{\mathfrak {s}}=\partial _z+f+\sum _{i=1}^n\frac{\bar{r}^{d_i}}{z^{d_i+1}}+\sum _{j \in J}\sum _{i=1}^{n} z^{-d_i}\sum _{l=0}^{d_i}\frac{s^{d_i}_l(j)}{(z-w_j)^{d_i+1-l}}, \end{aligned}$$

(6.2)

and the coefficients $\bar{r}^{d_i},s_l^{d_i}(j)\in \mathfrak {s}^{d_i}$ satisfy the conditions of Proposition 4.7. Since the term $z^{-h^{\vee }+1}(1+ \lambda z^{-\hat{k}})e_{\theta }$ is invariant under $\mathcal {N}(K_{\mathbb {P}^1})$, in order to prove our thesis we need to show that $\mathcal {L}_{\mathfrak {s}}$ is Gauge equivalent to

$$\begin{aligned} \mathcal {L}_1=\partial _z+f+\frac{r}{z}+\sum _{j \in J}\frac{1}{z-w_j}\left( -\theta ^\vee +\sum _{i=1}^{h^\vee -1} \frac{X^i(j)}{z^i} \right) , \end{aligned}$$

(6.3)

for some $X^i(j) \in \mathfrak {g}^i$. The proof of Theorem 6.1 is rather long but it is based on a simple principle. We show that the the canonical form of an oper of type $\mathcal {L}_1$ is an oper of type $\mathcal {L}_{\mathfrak {s}}$, and we prove that the induced map from the space of parameter of opers $\mathcal {L}_1$ to the space of parameters of opers $\mathcal {L}_{\mathfrak {s}}$ is bijective.

Remark 6.2

The choice of the representation (6.1) is motivated by the Bethe Ansatz equations. Indeed, according to Assumption 4 in order to construct solutions of the Bethe Ansatz equations, we need to impose on the additional singularities $w_j$ the trivial-monodromy conditions, which will result in a complete set of algebraic equations for the coefficients of the operator. Even though the location of the poles is independent of the choice of the Gauge, all local coefficients of course do depend on this choice. For theoretical and practical reasons we have chosen to work in the Gauge where all additional singularities are first order poles. Indeed, this Gauge does not depend on the choice of a transversal space, and the computation of the trivial monodromy conditions turns out to be much simpler. The computation of the monodromy at $z=w_j$ for opers of type (6.1) will be made in Sect. 9, where we will also show that the coefficients $X^i(j)\in \mathfrak {g}^i$ actually take values in the (symplectic) vector space $\mathfrak {t}\subset \mathfrak {b}_+$, which can be described as the orthogonal complement (with respect to the Killing form) of the subspace ${{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}{e_{-\theta }}$, where $e_{-\theta }$ is a lowest root vector of $\mathfrak {g}$.

Remark 6.3

Applying $z^{\rho ^\vee }$ to $\mathcal {L}_1$, we obtain

$$\begin{aligned} z^{\rho ^\vee } \mathcal {L}_1=\partial _z+\frac{f-\rho ^\vee +r}{z}+\sum _{j \in J}\sum _{i=0}^{h^\vee -1} \frac{X^i(j)}{z-w_j}. \end{aligned}$$

The (connection asssociated to the) above operator is totally Fuchsian: it is meromorphic on the Riemann sphere and all its singularities are first order poles. Hence, we can conclude that $\mathcal {L}_{\mathfrak {s}}$ is Gauge equivalent to a totally Fuchsian operator. We remark that the analysis of the similar question, namely whether a connection with only regular singularities is Gauge equivalent to a connection with only simple poles (i.e. a Fuchsian connection), is of primary importance in the theory of the Riemann–Hilbert problems and led to the negative solution of the Hilbert’s 21st problem, see [1].

7.1 Local theory of a Fuchsian singularity

The operator $\mathcal {L}_{\mathfrak {s}}$ given by (6.2) is fixed by the choice of the coefficients

$$\begin{aligned} s^{d_i}_l(j)\in \mathbb {C},\qquad 0\le l\le d_i,\,i \in I, \end{aligned}$$

(6.4)

namely by the choice of the singular coefficients of the Laurent expansion at any $w_j$. Similarly, the operator $\mathcal {L}_{\mathfrak {1}}$ given by (6.3) is fixed by the choice of

$$\begin{aligned} X^{i}(j)\in \mathfrak {g}^{i},\qquad i=0 \ldots h^{\vee }-1. \end{aligned}$$

(6.5)

Since the canonical form of $\mathcal {L}_1$ is $\mathcal {L}_{\mathfrak {s}}$, then the Gauge sending $\mathcal {L}_1$ to $\mathcal {L}_s$ induces a map from the parameters (6.5) to the parameters (6.4). This map is the object of our study. Our method of analysis is based on the reduction of the global problem to a simpler local one. This is the problem of proving that an operator of the form

$$\begin{aligned} \mathcal {L}=\partial _x+f+\sum _{i=1}^{h^{\vee }-1}\sum _{l=0}^{i}\frac{X^i_l}{x^{i+1-l}}, \end{aligned}$$

with given $X_l^i \in \mathfrak {g}^{i}$, is Gauge equivalent to an operator with a first order pole

$$\begin{aligned} \mathcal {L}=\partial _x+f+\sum _{i=0}^{h^{\vee }-1}\frac{X^i}{x}, \end{aligned}$$

for some $X^i \in \mathfrak {g}^i$. In order to do so, we describe the local structure of both operators and Gauge transformations at a Fuchsian singular point. We first embed the space ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ into a Lie algebra and then proceed with the localization.

Definition 6.4

Let D be a domain in $\mathbb {C}$. We denote

$$\begin{aligned} \mathfrak {g}'(K_D)=\mathfrak {g}(K_D)\oplus \mathbb {C}\partial _z \end{aligned}$$

the extension of $\mathfrak {g}(K_D)$ by the element $\partial _z$, with the relation $[\partial _z,p(z)]=\frac{d p}{dz}$.

It is clear that we have an injective map ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)\hookrightarrow \mathfrak {g}'(K_D)$.

Definition 6.5

Let $w \in \mathbb {C}$ and set $x=z-w$. We denote

$$\begin{aligned} \mathfrak {g}'((x))=\mathfrak {g}\otimes \mathbb {C}((x)) \oplus \mathbb {C}\partial _x \end{aligned}$$

(6.6)

the Lie algebra defined by the relations

$$\begin{aligned}{}[g_1\otimes x^m,g_2\otimes x^p ]=[g_1,g_2]\otimes x^{m+p},\qquad [\partial _x,g \otimes x^{m}]= g \otimes mx^{m-1} \end{aligned}$$

for $g_1,g_2,g\in \mathfrak {g}$ and $m,p\in \mathbb {Z}$.

We interpret the Lie algebra (6.6) as a localized version (at the point $x=z-w$) of operators in $\mathfrak {g}'(K_D)$, and in particular of operators in ${{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$. To make this statement more precise, we assign two different degrees to elements in $\mathfrak {g}'((x))$: the principal degree, given by

$$\begin{aligned} \deg \partial _x=0,\qquad \deg g^i \otimes x^j=i, \quad g^i \in \mathfrak {g}^i, \end{aligned}$$

and the total degree, given by

$$\begin{aligned} {{\,\mathrm{deg_{x}}\,}}\partial _x=-1,\qquad {{\,\mathrm{deg_{x}}\,}}g^i \otimes x^j=i+j,\quad g^i \in \mathfrak {g}^i. \end{aligned}$$

For every $k \in \mathbb {Z}$, we denote $\mathfrak {g}((x))^{\ge k} \subset \mathfrak {g}'((x))$ the subspace generated by elements with total degree greater than or equal to k. We also define the localized Gauge groups as

$$\begin{aligned} \mathcal {N}_{loc}&=\lbrace \exp {y}:\, y \in \mathfrak {n}_+\otimes \mathbb {C} ((x)) \rbrace ,\\ \mathcal {N}_{loc}^{\ge 0}&= \lbrace \exp {y}:\, y \in \mathfrak {n}_+\otimes \mathbb {C} ((x)) \cap \mathfrak {g}^{\ge 0}((x)) \rbrace . \end{aligned}$$

For $w\in D$, the localization map

$$\begin{aligned} Loc_w: \mathfrak {g}'(K_D) \rightarrow \mathfrak {g}'((x)), \end{aligned}$$

is defined by setting $Loc_w(\partial _z)=\partial _x$, and $Loc_w(g)$ to be the Laurent series at w of $g\in \mathfrak {g}(K_D)$. Fixed $w \in D$ the above map is an injective morphism of Lie algebras. We denote Loc(g) as $g_w$, for $g\in \mathfrak {g}'(K_D)$. By definition, if we localise $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ at a point w we obtain an element $\mathcal {L}_w$ of $\mathfrak {g}'((x))$, and if we localise a Gauge transformation $Y \in \mathcal {N}(K_D)$ at a point w we obtain an element $Y_w$ of $\mathcal {N}_{loc}$. Since the localisation map is a morphism then $(Y.\mathcal {L})_w=Y_w.\mathcal {L}_w$.

The total degree that we have introduced for the localisation of opers is useful to study fuchsian singularities. In fact, we have the following Lemma.

Lemma 6.6

• 1. Let $\mathcal {L} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$. Then $w \in D$ is a Fuchsian singularity of $\mathcal {L}$ if and only if $(\mathcal {L})_w \in \mathfrak {g}((x))^{\ge -1}$. If w is Fuchsian and $\mathcal {L}_{\mathfrak {s}}=Y.\mathcal {L}$ is the canonical form of $\mathcal {L}$, with $Y \in \mathcal {N}(K_D)$, then the localisation of Y at w belongs to $\mathcal {N}_{loc}^{\ge 0}$.

• 2. If $\mathcal {L},\widehat{\mathcal {L}} \in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$ are Gauge equivalent operators with a Fuchsian singularity at w, then the principal coefficients of the singularity are conjugated in $\mathcal {N}$.

Proof

• 1. Let $\mathcal {L}=\partial _z+f+b\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_D)$. By definition of Fuchsian singularity, then $z=w$ is Fuchsian if and only if

  $$\begin{aligned} (\mathcal {L})_w=\partial _x+f+\sum _{i\ge 0,m\ge 0} \frac{b^i_m}{x^{i+1-m}}, \end{aligned}$$

  for some $b^i_m\in \mathfrak {g}^i$. But since each summand has total degree $\ge -1$, this is precisely the condition that $(\mathcal {L})_w\in \mathfrak {g}^{\ge -1}((x))$. Now let w be a Fuchsian singularity for $\mathcal {L}$, and let $\mathcal {L}_s=Y\mathcal {L}$ be the canonical form of $\mathcal {L}$, for some $Y\in \mathcal {N}(K_D)$. We want to prove that $Y_w\in \mathcal {N}_{loc}^{\ge 0}$. Since w is Fuchsian for $\mathcal {L}$ then $(\mathcal {L})_w\in \mathfrak {g}((x))^{\ge -1}$, and due to Corollary 3.10w is Fuchsian also for $\mathcal {L}_{\mathfrak {s}}$, so that $(\mathcal {L}_{\mathfrak {s}})_w \in \mathfrak {g}((x))^{\ge -1}$. Note that by construction we have $Y_w\mathcal {L}_w=(Y\mathcal {L})_w=(\mathcal {L}_{\mathfrak {s}})_w$, from which we infer that $Y_w\mathcal {L}_w\in \mathfrak {g}((x))^{\ge -1}$. We prove that $Y_w\in \mathcal {N}_{loc}^{\ge 0}$ by showing that if $\overline{\mathcal {L}} \in \mathfrak {g}((x))^{\ge -1}$, and $Y\notin \mathcal {N}_{loc}^{\ge 0}$, then $Y \overline{\mathcal {L}} \notin \mathfrak {g}^{\ge -1}$. Indeed, let $Y=\exp {y}$ with $y\in \mathfrak {n}_+(K_D)$. Since $Y\notin \mathcal {N}_{loc}^{\ge 0}$, there exists a maximal $k>0$ such that the projection of y into the subspace of total degree $-k$ is non zero. Let then $\frac{y^i}{x^{i+k}}$, with $0\ne y^i \in \mathfrak {g}^i$ be the term of y of total degree $-k$ and of lowest principal degree i. Then the projection of $\exp {y}.\overline{\mathcal {L}}$ onto the subspace of total degree $-k-1$ is non trivial, as $[f,\frac{y^i}{x^{i+k}}]\ne 0$ is the unique term in $\exp {y}.\overline{\mathcal {L}}$ with total degree $-k-1$ and principal degree $i-1$. Hence, $Y \overline{\mathcal {L}} \notin \mathfrak {g}((x))^{\ge -1}$.

• 2. It is enough to prove the statement when $\widehat{\mathcal {L}}=\mathcal {L}_{\mathfrak {s}}$ is the canonical form of $\mathcal {L}$. Let $Y\in \mathcal {N}(K_D)$ be the Gauge transformation such that $\mathcal {L}_{\mathfrak {s}}=Y\mathcal {L}$. Since w is Fuchsian, then due to part (1) we have that $Y_w \in \mathcal {N}_{loc}^{\ge 0}$, and a direct calculation shows that $x^{{{\,\mathrm{ad}\,}}\rho ^\vee }Y_w$ is regular at $x=0$. In other words, $x^{{{\,\mathrm{ad}\,}}\rho ^\vee }Y_w=\exp {(\sum _{k \ge 0}y_k x^k)}$, for some $y_k \in \mathfrak {n}^+$. Again using the fact that w is Fuchsian, we obtain (cf. equation (3.3)) that $x^{{{\,\mathrm{ad}\,}}\rho ^\vee }\mathcal {L}_w=\partial _x+\frac{a}{x}+O(1)$ and $x^{{{\,\mathrm{ad}\,}}\rho ^\vee }(\mathcal {L}_{\mathfrak {s}})_w=\partial _x+\frac{b}{x}+O(1)$, where $a,b\in \mathfrak {g}$ are the principal coefficients of $\mathcal {L}$ and $\mathcal {L}_{\mathfrak {s}}$ respectively. Since $(\mathcal {L}_{\mathfrak {s}})_w=Y_w\mathcal {L}_w$ and $x^{{{\,\mathrm{ad}\,}}\rho ^\vee }Y_w$ is regular at $x=0$, we obtain the relation $b=\exp {y_0}.a$. $\quad \square$

We now introduce three important classes of operators in $\mathfrak {g}((x))^{\ge -1}$.

Definition 6.7

We say that $\mathcal {L}$ is a $\mathfrak {g}$-Bessel (or simply a Bessel) operator if

$$\begin{aligned} \mathcal {L}=\partial _x+f+\sum _{i=0}^{h^{\vee }-1}\sum _{l=0}^{i}\frac{X^i_l}{x^{i+1-l}}, \qquad X_l^i \in \mathfrak {g}^{i}. \end{aligned}$$

(6.7)

Given a transversal space $\mathfrak {s}$, we say that $\mathcal {L}$ is a $\mathfrak {s}$-Bessel operator if

$$\begin{aligned} \mathcal {L}=\partial _x+f+\sum _{i=1}^{n}\sum _{l=0}^{d_i}\frac{s^{d_i}_l}{x^{d_i+1-l}},\qquad s_l^{d_i} \in \mathfrak {s}^{d_i}. \end{aligned}$$

(6.8)

We denote by $V=V_{\mathfrak {s}}$ the affine vector space of $\mathfrak {s}$-Bessel operators. We say that $\mathcal {L}$ is a 1-Bessel operator if

$$\begin{aligned} \mathcal {L}=\partial _x+f+ \frac{X}{x},\qquad X\in \mathfrak {b}_+. \end{aligned}$$

(6.9)

We denote by U the affine vector space of 1-Bessel operators.

In the case $\mathfrak {g}=\mathfrak {sl}_2$, Bessel operators coincide with the operators of the Bessel differential equation. As shown below, the canonical form of every Bessel operator is an $\mathfrak {s}$-Bessel operator.

Lemma 6.8

Any Bessel operator (6.7) is Gauge equivalent to an $\mathfrak {s}$-Bessel operator (6.8). The corresponding Gauge transformation belongs to the finite dimensional subgroup $\overline{\mathcal {N}}_{loc}\subset \mathcal {N}_{loc}^{\ge 0}$ generated by elements in $\mathfrak {n}_+((x))$ without regular terms

$$\begin{aligned} \overline{\mathcal {N}}_{loc}=\lbrace \exp {y}\,: \, y =\sum _{i=1}^{h^{\vee }-1}\sum _{j=1}^{i}\frac{y^i_j}{x^j}, \, y^i_j \in \mathfrak {g}^i \rbrace . \end{aligned}$$

Proof

A simple computation shows that the set of Bessel operators is invariant under the action of the group $\overline{\mathcal {N}}_{loc}$. We can then prove the Lemma using the same steps as in the proof of Proposition 2.8. We factorize $Y\in \overline{\mathcal {N}}_{loc}$ as follows: $Y=Y_{h^{\vee }-2} \ldots Y_1 Y_0$ where $Y_j= Y^{h^{\vee }-1}_j \ldots Y^{j+1}_j$ and $Y^i_j=\exp {\frac{y^{i}_j}{x^{i-j}}}$ for some $y^i_j \in \mathfrak {g}^i$. The transformation $Y^i_j=\exp {\frac{y^{i}_j}{x^{i-j}}}$ is then defined by recursively imposing that, after its application, the terms of the resulting operator with total degree $\le j-1$ and principal degree $\le i-1$ are in canonical form. $\quad \square$

Due to the previous lemma, we have a well-defined map from Bessel operators to $\mathfrak {s}$-Bessel operators. We are interested in the restriction of this map to the class of 1-Bessel operators. Note that once further restricted to the class of 1-Bessel operators of the form $\partial _x+f+X^0/x$, with $X^0\in \mathfrak {g}^0=\mathfrak {h},$ then this map should be thought as a local version, at a regular singular point, of the so-called ‘Miura map’. Bessel operators will play a prominent role later in this section, to obtain a normal form for Quantum $\widehat{\mathfrak {g}}$-KdV opers.

The space U of 1-Bessel operators (6.9) can be described by means of the graded affine space

$$\begin{aligned} U=\bigoplus _{i=0}^{h^{\vee }-1} U_i, \end{aligned}$$

where

$$\begin{aligned} U_0=\lbrace \partial _x+f+x^{-1} X^0\,|\, X^0 \in \mathfrak {h} \rbrace ,\qquad U_i=x^{-1}\mathfrak {g}^i,\quad i>0. \end{aligned}$$

Note that ${{\,\mathrm{deg_{x}}\,}}U_i=i-1$. In the sequel we will often identify $U_i$ with $\mathfrak {g}^i$ and $\oplus _{i\ge 1}U_i$ with $\mathfrak {n}^+$. Similarly, the space V of $\mathfrak {s}$-Bessel operators (6.8) can be written as

$$\begin{aligned} V=\bigoplus _{i=0}^{h^{\vee }-1} V_{i}, \end{aligned}$$

where

$$\begin{aligned} V_{0}=\lbrace \partial _x+f+\sum _{i=1}^n \frac{s^{d_i}}{x^{d_i+1}} \,|\, s^{d_i} \in \mathfrak {s}^{d_i} \rbrace ,\qquad V_{i}= \lbrace \sum _{d_j\ge i}\frac{s^{d_j}}{x^{d_j+1-i}}\,|\, s^{d_j} \in \mathfrak {s}^{d_j}\rbrace , \quad i>0. \end{aligned}$$

Note that ${{\,\mathrm{deg_{x}}\,}}{V_{i}}=i-1$.

Lemma 6.9

The space U of 1-Bessel operators and the space V of $\mathfrak {s}$-Bessel operators have the same dimension. More precisely, $\dim U=\dim V=\dim \mathfrak {b}_+=(\frac{h^{\vee }}{2}+1)n$.

Proof

It is clear that $U\simeq \mathfrak {b}_+$, so in particular $\dim U=\dim \mathfrak {b}_+=(\frac{h^{\vee }}{2}+1)n$. We prove by induction on i that $U_i$ and $V_i$ have the same dimension. First, $U_0\cong \mathfrak {h}$ and $V_0\cong \mathfrak {s}$, so $\dim U_0=\dim \mathfrak {h}=n=\dim \mathfrak {s}=\dim V_0$. Then, it is clear from the definition of $V^i$ that $\dim V^{i}=\dim V^{i+1}+\dim \mathfrak {s}^i$, where $\mathfrak {s}^i=\mathfrak {s}\cap \mathfrak {g}^i$, and by definition of transversal space we have $\dim \mathfrak {s}^i=\dim \mathfrak {g}^{i}-\dim \mathfrak {g}^{i+1}$. Since $U^i\simeq \mathfrak {g}^i$, then we have $\dim V^{i}-\dim V^{i+1}=\dim \mathfrak {s}^i=\dim \mathfrak {g}^{i}-\dim \mathfrak {g}^{i+1}=\dim U^{i}-\dim U^{i+1}$. Hence $\dim U^{i}=\dim V^i$ implies $\dim U^{i+1}=\dim V^{i+1}$. $\quad \square$

Definition 6.10

We denote as $\Phi : U \rightarrow V$ be the map that associates to any 1-Bessel operator its canonical form. We define $\Phi _i$ as the projection of $\Phi$ onto $V_i$, so that the decomposition $\Phi =\oplus _i \Phi _i$ holds true.

Remark 6.11

Let $\mathcal {L}=\partial _x+f+x^{-1}X\in U$ be a 1-Bessel operator, with $X\in \mathfrak {b}_+$. Let $X=\sum _{i\ge 0}X^i$ be the decomposition of X according to the principal gradation, with $X^i\in \mathfrak {g}^i$. By abuse of notation, we write $\Phi (X^0,\ldots ,X^{h^\vee -1})$ to denote $\Phi (\mathcal {L})$.

After Lemma 6.8, the Gauge transformation N mapping a 1-Bessel operator to its canonical form belongs to $\overline{\mathcal {N}}_{loc}$. In particular, $Y=\exp y$ with $y\in \mathfrak {g}((x))^{\ge 0}$, that is a linear combination of terms of non-negative total degree. It follows form this that, for each i, the map $\Phi _i$ depends only on $\oplus _{j\le i}U_j$. More precisely, we have

Lemma 6.12

Let $\mathcal {L}=\partial _x+f+x^{-1}X\in U$, with $X\in \mathfrak {b}_+$, be a 1-Bessel operator, and let $X=\sum _{i\ge 0}X^i$, with $X^i\in \mathfrak {g}^i$. The map $\Phi :U\rightarrow V$ which associates to $\mathcal {L}$ its canonical form $\mathcal {L}_{\mathfrak {s}}$ admits the triangular decomposition:

$$\begin{aligned} \Phi (X^0,\ldots ,X^{h^{\vee }-1})=\sum _{i=0}^{h^{\vee }-1} \Phi _{i}(X^0,\ldots ,X^i), \qquad \Phi _i: \bigoplus _{j\le i}U_j \rightarrow V_i. \end{aligned}$$

(6.10)

In other words, the terms of total degree $i-1$ in $\mathcal {L}_{\mathfrak {s}}$ depend on the terms of total degree $\le i-1$ of $\mathcal {L}$ only.

Proof

We prove the following equivalent statement: if $\mathcal {L}=\mathcal {L}'$ up to terms of total order $\le i-1$, then the canonical form of $\mathcal {L}$ coincides up to terms of total order $\le i-1$ with the canonical form of $\mathcal {L}'$.

Let $\mathcal {L}$ be a 1-Bessel operator and $Y=Y_{h^{\vee }-2}...Y_0$ be the corresponding Gauge transformation—factorized as in the proof of Lemma 6.8—mapping $\mathcal {L}$ to its canonical form $\mathcal {L}_{\mathfrak {s}}$. If $\mathcal {L}'$ coincides with $\mathcal {L}$ for terms of total degree $\le i-1$, then the terms of total degree $\le i-1$ of $Y_i \ldots Y_0 \mathcal {L}'$ and $Y_i \ldots Y_0 \mathcal {L}$ are also the same. By construction, $Y_i \ldots Y_0 \mathcal {L}$ coincides with its canonical form $\mathcal {L}_{\mathfrak {s}}$ up to terms of total degree $\le i-1$. It follows that $Y_i \ldots Y_0 \mathcal {L}'$ is in canonical form except that for terms of total degree $\ge i$. Hence, the transformation $Y'$ mapping $\mathcal {L}'$ to its canonical form can be factorised as $Y'=Y'_{h^{\vee }-2}\ldots Y'_{i+1}Y_i \ldots Y_0$ where $Y'_j=\exp y_j$, with $j \ge i+1$, and ${{\,\mathrm{deg_{x}}\,}}y_j=j$. It follows that $Y_i \ldots Y_0 \mathcal {L}'$ coincides with the canonical form of $\mathcal {L}'$ up to terms of total degree $\le i-1$. $\quad \square$

7.2 The maps ${\Phi _i, i\ge 0}$

In order to study the properties (in particular the surjectivity) of the map $\Phi$, we first study the map $\Phi _0$, and then the maps $\Phi _i, i\ge 1$. We begin with the following

Lemma 6.13

Let $f+\mathfrak {s}$ be a transversal space and $h \in \mathfrak {h}$ an element of the Cartan subalgebra. Then $f+h+\mathfrak {s}$ is a transversal space too.

Proof

Since $\mathfrak {s}$ is transversal, every $b\in \mathfrak {b}_+$ can be written as $b=[f,m]+s$, for some $m\in \mathfrak {n}_+$ and $s\in \mathfrak {s}$. We prove that there exists an $y \in \mathfrak {n}^+$ such that $b=[f,y]+h+s$. Since ${{\,\mathrm{ad}\,}}f_{|\mathfrak {g}^1}: \mathfrak {g}^1 \rightarrow \mathfrak {h}$ is invertible, we denote by $g\in \mathfrak {g}^1$ the unique element such that $[f,g]=h$. Setting $y=m-g$, then $b=[f,y]+h+s$. $\quad \square$

We next prove that the map $\Phi _0:U_0 \cong \mathfrak {h} \rightarrow V_0\cong \mathfrak {s}$ is surjective and it is invariant under the dotted Weyl group $\mathcal {W}$ action on $\mathfrak {h}$. The latter is defined as

$$\begin{aligned} \sigma \cdot h= \sigma (h-\rho ^\vee )+\rho ^\vee , \end{aligned}$$

(6.11)

for $\sigma \in \mathcal {W}$ and $h \in \mathfrak {h}$.⁷

Lemma 6.14

Let $\mathcal {L}_0=\partial _x+f+ x^{-1}X^0$ be a 1-Bessel operator, with $X^0\in \mathfrak {h}$, and fix a trasversal subspace $\mathfrak {s}$. Let $(s,y_0)\in \mathfrak {s}\times \mathfrak {n}_+$ be the unique pair of elements such that $\exp (y_0).(f-\rho ^\vee +X^0)=f-\rho ^\vee +s$.

• 1. The canonical form of $\mathcal {L}_0$ is

  $$\begin{aligned} \mathcal {L}_{\mathfrak {s}}=\partial _x+f+\sum _{i=1}^n\frac{s^{d_i}}{x^{d_i+1}}, \end{aligned}$$

  where $s^{d_i}\in \mathfrak {s}^{d_i}$ is the restriction of s to $\mathfrak {s}^{d_i}$.

• 2. Let $y_0=\sum _i y_0^i$, with $y_0^i\in \mathfrak {g}^i$, and let $Y_0=\exp \sum _{i}\frac{y_0^i}{x^i}\in \overline{\mathcal {N}}_{loc}$. Then $\mathcal {L}_{\mathfrak {s}}=Y_0.\mathcal {L}_0$. In particular, ${{\,\mathrm{deg_{x}}\,}}\sum _{i}\frac{y_0^i}{x^i}=0$.

• 3. The map

  $$\begin{aligned} \Phi _0: U_0 \cong \mathfrak {h} \rightarrow V_0,\qquad \Phi _0(X^0)=s \end{aligned}$$

  is surjective.

• 4. $\Phi _0(h)=\Phi _0(h')$ if and only if there exists $\sigma \in \mathcal {W}$ such that $\sigma \cdot h=h'$.

Proof

A proof of (1) and (2) is already contained in the proof of Proposition 4.1. We give here another proof, more algebraic in nature. Fix $X^0 \in \mathfrak {h}$ and consider the operator $\mathcal {L}=\partial _x+f+x^{-1}X^0$. Since $f-\rho ^\vee +\mathfrak {s}$ is a transversal space then the map $\mathcal {N} \times \mathfrak {s} \rightarrow f+\mathfrak {b}^+$, $(Y,s) \mapsto Y.(f-\rho ^\vee +s)$ is an isomorphism of affine varieties [35]; in particular given $X^0$ there exists a unique pair $(s,y_0) \in \mathfrak {s} \times \mathfrak {n}^+$ such that $\exp {y_0}.(f-\rho ^\vee +X^0)=f-\rho ^\vee +s$. Hence,

$$\begin{aligned} \exp {y_0}.\big (\partial _x+ \frac{f-\rho ^\vee +X^0}{x}\big )=\partial _x+ \frac{f-\rho ^\vee +s}{x}. \end{aligned}$$

From this, it follows that the Gauge $Y_0=x^{-{{\,\mathrm{ad}\,}}\rho ^\vee }\exp {y_0}=\exp {\sum _i \frac{y_0^i}{x^i}}$ maps $\partial _x+f+ x^{-1}X^0$ to $\partial _x+f+\sum _{i} x^{-d_i-1}s^{d_i}$, where $s^{d_i}$ is the projection of s onto $\mathfrak {s}^{d_i}$.

(3,4) Recall that a transversal space is in bijection with the regular G orbits. Hence (3) $\Phi _0$ is surjective if and only if every regular G orbit intersects the affine space $f+\mathfrak {h}$, and (4) $\Phi _0(h)=\Phi _0(h')$ if and only if $f-\rho ^\vee +h$ and $f-\rho ^\vee +h'$ belong to the same G orbit. It is proved in [35] that every regular G intersects $f+\mathfrak {h}$ and two elements $f+l,f+l'$, $l,l'\in \mathfrak {h}$ belong to the same G orbit if and only if l and $l'$ belong to the same W orbit. $\quad \square$

We now turn our attention to the maps $\Phi _i\cup _{j\le i}U_i \rightarrow V_i$, $i\ge 1$.

Lemma 6.15

Let $i\ge 1$. For every $X_0 \in \mathfrak {h}$, there exists a linear map $A_i^{X_0}:\mathfrak {g}^i \rightarrow V_i$ such that

$$\begin{aligned} \Phi _i(X^0,\ldots ,X^i)=A^{X_0}_i[X^i]+P_i(X^0,\ldots ,X^{i-1}) \end{aligned}$$

(6.12)

for some $P_i:\cup _{j\le i-1}U_j \rightarrow V_i$.

Proof

Let $\mathcal {L}\in U$ be a 1-Bessel operator, of the form $\mathcal {L}=\partial _x+f+x^{-1}X$ with $X\in \mathfrak {b}_+$. Let $X=\sum _{i\ge 0}X^i$, with $X^i\in \mathfrak {g}^i$ and for $i\ge 0$ denote

$$\begin{aligned} \mathcal {L}_i=\partial _x+f+\frac{1}{x}\sum _{j=0}^{i}X^j. \end{aligned}$$

(6.13)

By Lemma 6.8, in order to reconstruct $\Phi _i$ is is enough to find a Gauge transformation $Y_i$ such that $\Pi _f\big (Y_i \mathcal {L}_i-\partial _x \big )$—the projection of $Y_i \mathcal {L}_i-\partial _x$ onto the space $[f,\mathfrak {n}_+]$—is a linear combination of terms of total degree $\ge i$. Indeed, this condition is satisfied if and only if the operator $Y_i\mathcal {L}_i$ is canonical up to total degree $i-1$, in which case the projection of $Y_i\mathcal {L}_i$ onto $V_i$ coincides—by Lemma 6.8—with the map $\Phi _i$.

For $i=0$, then $\mathcal {L}_0=\partial _x+f+x^{-1}X^0$ with $X^0\in \mathfrak {h}$, and the Gauge $Y_0$ was obtained in Lemma 6.14. We now construct $Y_i$ recursively with respect to the total gradation as $Y_i=\exp {y_i}Y_{i-1}$, where

$$\begin{aligned} y_i(x)=\sum _{j= i}^{h^{\vee }-1}\frac{y_i^{j}}{x^{j-i}} \end{aligned}$$

(6.14)

is an element of the loop algebra of total degree i. Notice that if we let $\mathcal {L}$ vary as a function of the variables $X^0,\ldots X^{h^{\vee }-1}$, then the transformation $Y_i$ is a function of $X^0,\ldots X^i$ only. Since $\Phi _j(X^0,\ldots ,X^{j-1})\in V_i$, each map $\Phi _i$ admits the decomposition

$$\begin{aligned} \Phi _j(X^0,\ldots ,X^{j-1})=\sum _{\ell \ge j}\frac{\Phi ^\ell _j}{x^{\ell +1-j}}, \end{aligned}$$

(6.15)

for some $\Phi ^\ell _j\in \mathfrak {s}^{\ell }$. Now assume that for $j\le i-1$ the Gauge $Y_{j}$ and the maps $\Phi _j(X^0,\ldots ,X^j)$ have been obtained, so that by construction the projection $\Pi _f(Y_{i-1}\mathcal {L}_{i-1})$ contains elements of total degree $\ge i-1$ only. By construction, the operator $Y_{i-1} \mathcal {L}_{i-1}$ has the form

$$\begin{aligned} Y_{i-1} \mathcal {L}_{i-1}=\partial _x+f+\sum _{j\le i-1}\sum _{\ell \ge j}\frac{\Phi ^\ell _{j}}{x^{\ell +1-j}}+ \sum _{j\ge i-1} b_j(x), \end{aligned}$$

where $b_j(x)=\sum _{l\ge 0} \frac{b^l_j}{x^{l-j}}$, with $b^j_l\in \mathfrak {g}^j$, is a remainder term of of total degree $j\ge i-1$. We now look for an element $y_i$ of the form (6.14) such that $\Pi _f\big (\exp {y_i} Y_{i-1} \mathcal {L}_i \big )$ contains only elements of total degree $\ge i$, thus proving the induction step. Due to (6.13) we have

$$\begin{aligned} \exp y_i.Y_{i-1}\mathcal {L}_i=\exp y_i.Y_{i-1}(\mathcal {L}_{i-1}+x^{-1}X^i), \end{aligned}$$

and since $y_i$ is of total degree i, it follows that the terms of total degree $\le i-2$ are already in canonical form. This is equivalent to say that $\Pi _f\big (\exp {y_i} Y_{i-1} \mathcal {L}_i \big )$ contains elements of total degree $\ge i-1$ only. It remains to consider the terms in $\exp {y_i} Y_{i-1} \mathcal {L}_i$ of total degree equal to $i-1$. These are given by

$$\begin{aligned} b_{i-1}(x)+\frac{1}{x}Y_0.X^i+[y_i(x),\partial _x+f+\sum _{\ell \ge 0}\frac{\Phi _0^\ell }{x^{\ell +1}}], \end{aligned}$$

and the required condition is obtained imposing that the above quantity belongs to $V_i$. Due to the definition of the map $\Phi$, this is equivalent to say that

$$\begin{aligned} \Phi _i(X^0,\ldots ,X^{i-1})=b_{i-1}(x)+\frac{1}{x}Y_0.X^i+[y_i(x),\partial _x+f+\sum _{\ell \ge 0}\frac{\Phi _0^\ell }{x^{\ell +1}}]. \end{aligned}$$

The above is a system of equations for the coefficients $\Phi _i^j\in \mathfrak {g}^j$, related to $\Phi _i$ by (6.15) and $y^{j}_i\in \mathfrak {g}^{j}$, related to $y_i(x)$ by (6.14). Applying the Gauge $x^{{{\,\mathrm{ad}\,}}\rho ^\vee }$ on both sides, we obtain the following set of equations in $\mathfrak {g}$:

$$\begin{aligned} \sum _{j\ge i}(\Phi ^j_i+[f,y_i^{j+1}])=\sum _{j\ge 0} b^j_{i-1}+\exp y_0.X^i+\sum _{j\ge i}(j-i) y^{j}_i-[\sum _{j\ge i}y^{j}_i,\sum _{\ell \ge 0}\Phi _0^\ell ], \end{aligned}$$

where $y_0\in \mathfrak {n}_+$ is the element obtained in Lemma 6.14. Decomposing according to the principal gradation and projecting onto the subspaces $[f,\mathfrak {n}_+]$ and $\mathfrak {s}$, we obtain the following system for the elements $y_i^{j}$ and $\Phi ^j_i$, with $j\ge i$:

$$\begin{aligned}&\Phi ^{j}_i=\Pi _{\mathfrak {s}}\big ((j-i) y^{j}+b^{j}_{i-1}+\big (\exp y_0. X^i\big )^{j}+ \sum _{m+l=j} [y^{m},\Phi ^{l}_{0}]\big ), \end{aligned}$$

(6.16a)

$$\begin{aligned}&[f,y^{j+1}]=\Pi _f \big ((j-i) y^{j}+b^{j}_{i-1}+\big (\exp y_0.X^i\big )^{j}+ \sum _{m+l=j} [y^{m},\Phi ^{l}_0]\big ), \end{aligned}$$

(6.16b)

where $\big (\exp y_0. X^i\big )^{j}$ denotes the projection of $\exp y_0. X^i$ onto $\mathfrak {g}^j$. The system has a unique solution since $({{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}_f ) \cap \mathfrak {n}^+ =0$.

We can now study how the map $\Phi _i$ depends on the variables $X_0,\ldots ,X_i$ when we let $\mathcal {L}$ vary, in order to prove the decomposition (6.12). By construction, the quantity $b^{j}_{i-1}$, depends on $X^0,\ldots X^{i-1}$ only. In addition, since $\exp y_0$ and $\Phi _0$ depend on $X^0$ only, then the quantity $\tilde{\Phi }^{j}_i:=\Phi ^{j}_i -\Pi _{\mathfrak {s}}(b^{j}_{i-1})$ depend exclusively on $X^0$ and $X^i$. Moreover, it depends linearly on the variable $X^i$. Indeed, both $\tilde{\Phi }^{i}_i$ and $y^{i+1}_i$ are linear in $X^i$, and at each subsequent steps $\tilde{\Phi }^{j}_i$ and $y^{j}_i$ depend linearly on the previous $\tilde{\Phi }$’s and y’s. This proves the thesis. $\quad \square$

We now consider the behaviour of the map $\Phi (X^0,\ldots ,X^{h^\vee -1})$ for fixed values of the first entry $X^0\in \mathfrak {h}$.

Definition 6.16

Fixed $X^0 \in \mathfrak {h}$, we denote $\Phi ^{X^0}:\mathfrak {n}^+ \rightarrow \bigoplus _{i\ge 1} V_i,$ the map

$$\begin{aligned} (X^1,\ldots X^{h^{\vee }-1})\mapsto \sum _{i\ge 1}\Phi _i(X^0,X^1,\ldots X^{i}), \end{aligned}$$

so that the decomposition

$$\begin{aligned} \Phi (X^0,\ldots ,X^{h^{\vee }-1})=\Phi _0(X^0)+\Phi ^{X^0}(X^1,\ldots ,X^{h^{\vee }-1}) \end{aligned}$$

holds true.

Proposition 6.17

• 1. The map $\Phi ^{X^0}$ is injective if and only if it is surjective.

• 2. If the map $\Phi ^{X_0}:\mathfrak {n}_+ \rightarrow \sum _{i\ge 1}V_i$ fails to be surjective then there exists an $i\ge 2$ and a non-zero element $y \in \mathfrak {g}^i$ such that $[X_0,y]= y$.

• 2. There exists an open and dense subset $\mathcal {A} \subset \mathfrak {h}$ such that the map $\Phi ^{X^0}$ is surjective and injective for all $X^0 \in \mathcal {A}$

Proof

• 1. Due to the triangular decomposition (6.10) and to Lemma 6.15, the map $\Phi ^{X^0}$ is surjective if and only if $\Phi _i^{X_0}$ is surjective for all $i\ge 1$, which is equivalent to the condition $\det A_i^{X_0} \ne 0$ for all $i\ge 1$, which is equivalent to the condition $\Phi _i$ is injective for all $i\ge 1$, which is equivalent to the condition that $\Phi ^{X^0}$ is injective.

• 2. Let $U_{X^0}$ be the affine space $\partial _x+f+x^{-1}(X^0+Y)$ with $Y \in \mathfrak {n^+}$. If $\Phi ^{X^0}$ is not surjective, then by part (1) it is not injective. Hence, there exist operators $\mathcal {L},\overline{\mathcal {L}} \in U_{X^0}$ and (due to Lemma 6.8) Gauge transformations $M,\bar{M} \in \overline{\mathcal {N}}_{loc}$ such that $M\mathcal {L}=\bar{M}\overline{\mathcal {L}}$. Thus, $Y=\bar{M}^{-1}M \in \overline{\mathcal {N}}_{loc}$ satisfies $Y\mathcal {L}=\overline{\mathcal {L}}$. Since $Y\in \overline{\mathcal {N}}_{loc}$, we take it to be of the form $Y=\exp \sum _{i=1}^{h^{\vee }-1}\sum _{j=1}^i\frac{y_j^i}{x^j}$ with $y^i_j \in \mathfrak {g}^i$. Now let $I\ge 1$ the minimal index i such that $y^i_j\ne 0$ for at least one $1\le j\le i$. Then, a direct calculation shows that the only term of principal degree $I-1$ in $\mathcal {L}-\overline{\mathcal {L}}$ is given by

  $$\begin{aligned} q^I=\sum _{j}[f,\frac{y^I_j}{x^j}] \in \mathfrak {g}^{I-1}, \end{aligned}$$

  (6.17)

  Since $\mathcal {L},\overline{\mathcal {L}} \in U_{X^0}$, then the terms of total degree $<0$ coincide, from which it follows that necessarily $q^I$ is of total degree greater than 0. By looking at (6.17) we thus obtain that $q^I=O(\frac{1}{x})$, from which we deduce that $I\ge 2$ and $y^{I}_j=0$ for all $j\ge 2$. Collecting the terms of principal degree I in $\mathcal {L}-\overline{\mathcal {L}}$ according to formula (2.5), we get

  $$\begin{aligned} \frac{[y^I_1,X_0]}{x^{2}}+\frac{ y^I_1}{x^{2}}. \end{aligned}$$

  Since $Y\mathcal {L}=\overline{\mathcal {L}}$ belongs to $U_{X_0}$, the above term must vanish. Then, the non-zero element $y^I_1\in \mathfrak {g}^I$, $I\ge 2$, satisfies $[X_0,y^I_1]=y^I_1$.

• 3. Due to (2), the set of $X^0$ such that $\Phi ^{X^0}$ is not bijective has positive codimension. $\quad \square$

The following result, which is of crucial importance in our construction, is a straightforward corollary of the previous Proposition.

Corollary 6.18

The map $\Phi ^{-\theta ^\vee }:\mathfrak {n}_+ \rightarrow \sum _{i\ge 1}V_i$ is surjective.

Proof

The spectrum of ${{\,\mathrm{ad}\,}}_{-\theta _0^\vee }$ restricted to $\mathfrak {n}^+$ does not contain positive integers: indeed it is $\lbrace 0,-1,-2\rbrace$ if $\mathfrak {g}\ne A_1$, and $\lbrace 0,-2\rbrace$ otherwise. $\quad \square$

Example 6.19

The case $\mathfrak {g}=A_1$. In this case $\mathfrak {n}^+=\mathfrak {g}^{1}$. Since $\mathfrak {g}^{j}$ with $j\ge 2$ is empty, then - by Proposition 6.17(2)- the map $\Phi ^{X^0}$ is bijective for every $X^0 \in \mathfrak {h}$.

Example 6.20

The case $\mathfrak {g}=A_2$. We have that $\mathfrak {n}^+=\mathfrak {g}^1\oplus \mathfrak {g}^2$, with $\mathfrak {g}^2=\mathbb {C} e_\theta$. Given $X^0\in \mathfrak {h}$, we show that the map $\Phi ^{X^0}: \mathfrak {n}^+\rightarrow V_1 \oplus V_2$ fails to be surjective if and only if $[X^0,e_\theta ]=e_\theta$. We deduce that, for this particular Lie algebra, the necessary condition described in Proposition 6.17 (2), for $\Phi ^{X^0}$ not being injective, is also sufficient. Consider the Gauge $Y=\exp {\frac{e_\theta }{x}}$. If $\mathcal {L}=\partial _x+f+\frac{X^0+X}{x}$, with $X \in \mathfrak {n}_+$ is a 1-Bessel operator, then $\overline{\mathcal {L}}=Y.\mathcal {L}=\mathcal {L}+\frac{ [f,e_\theta ]}{x}+\frac{e_\theta +[e_\theta ,X^0]}{x^2}$. So, $\overline{\mathcal {L}}$ is a 1-Bessel operator (namely, it belongs to the domain of $\Phi _0$) if and only if $X^0$ satisfies $[X^0,e_\theta ]=e_\theta$. If this is the case, then $\overline{\mathcal {L}}\ne \mathcal {L}$ but $\Phi ^{X^0}(\mathcal {L})=\Phi ^{X^0}(\overline{\mathcal {L}})$. Hence $\Phi ^{X^0}$ is not injective, nor surjective.

7.3 Extended Miura map for Quantum-KdV opers

Building on the theory of Bessel operators described above, we address here the main topic of the present section. The problem is to find a Gauge transformation mapping the oper

$$\begin{aligned} \mathcal {L}_{\mathfrak {s}}=\partial _z+f+\sum _{i=1}^n \frac{\bar{r}^{d_i}}{z^{d_i+1}}+\sum _{j \in J}\sum _{i=1}^{n} z^{-d_i}\sum _{l=0}^{d_i}\frac{s^{d_i}_l(j)}{(z-w_j)^{d_i+1-l}}, \end{aligned}$$

(6.18)

with $\bar{r}^{d_i}, s^{d_i}(j) \in \mathfrak {s}^{d_i}$, to an operator of the form

$$\begin{aligned} \mathcal {L}_1=\partial _z+f+\frac{r}{z}+\sum _{j \in J}\frac{1}{z-w_j}\sum _{i=0}^{h^\vee -1} \frac{X^i(j)}{z^i}, \end{aligned}$$

(6.19)

with $r \in \mathfrak {h}$ and $X^i(j) \in \mathfrak {g}^i$. As a first step, we prove that the canonical form of the operator (6.19) is of type (6.18).

Lemma 6.21

For an arbitrary choice of its parameters, the canonical form of the operator $\mathcal {L}_1$ (6.19) is an operator of the form $\mathcal {L}_{\mathfrak {s}}$ (6.18). Moreover, one has that $\sum _{i=1}^n \bar{r}^{d_i}=\Phi _0(r)$ and $\sum _i^n \frac{s^{d_i}_0(j)}{w_j^{d_i}}=\Phi _0(X^0(j)),$ for $j \in J$.

Proof

The operator $\mathcal {L}_1$ satisfies the following property: it is regular outside $0,\infty$ and $w_j,j \in J$, and these points are at most fuchsian singularities. From Corollary 3.10, the canonical form of $\mathcal {L}_1$ satisfies the same property. To prove the first part of the Lemma, it is then sufficient to show that any operator in canonical form with such a property is an operator of the form (6.18).

So let $\mathcal {L}_{\mathfrak {s}}=\partial _z+f+s$, for some $s\in \mathfrak {s}(K_{\mathbb {P}^1})$. From the definition of Fuchsian singularity it follows that if $0,\infty$ are Fuchsian then s must satisfy $s^{d_i}=O(z^{-d_i-1})$ as $z \rightarrow 0$, and $s^{d_i}=O(z^{-d_i-1})$ as $z \rightarrow \infty$. Now let

$$\begin{aligned} \bar{r}^{d_i}=\lim _{z \rightarrow 0}z^{d_i+1}s^{d_i}(z),\qquad \hat{s}(z)=s(z)-\sum _{i=1}^n \frac{\bar{r}^{d_i}}{z^{d_i+1}}. \end{aligned}$$

One has that $\hat{s}(z)=O(z^{-d_i})$ as $z \rightarrow 0$, and $\hat{s}(z)=O(z^{-d_i+1})$ as $z \rightarrow \infty$. Due to Lemma 4.6, the function $\hat{s}$ admits the decomposition

$$\begin{aligned} \hat{s}(z)=\sum _{j \in J}\sum _{i=1}^{n}z^{-d_i}\sum _{l=0}^{m_i(j)}\frac{s^{d_i}_l(j)}{(z-w_j)^{m_i(j)+1-l}}, \end{aligned}$$

for some $m_i(j)$. Since $w_j$ is Fuchsian for every $j \in J$, then $m_i(j)=d_i$. Hence, $\mathcal {L}_{\mathfrak {s}}$ is of the form (6.18).

The principal coefficient of $\mathcal {L}_{\mathfrak {s}}$ at 0 (resp. $w_j$) is $\sum _{i=1}^n \bar{r}^{d_i}$ (resp. $\sum _i\frac{s^{d_i}_0(j)}{w_j^{d_i}}$). The principal coefficient of $\mathcal {L}_{1}$ at 0 (resp. $w_j$) is r (resp. $X_0^{(j)}$). By Lemma 6.6(ii), the principal coefficients of $\mathcal {L}_1$ and $\mathcal {L}_{\mathfrak {s}}$ at any singularity must be conjugated. This is equivalent—see Lemma 6.14—to the conditions $\sum _{i=1}^n \bar{r}^{d_i}=\Phi _0(r)$ and $\sum _i\frac{s^{d_i}_0(j)}{w_j^{d_i}}=\Phi _0(X^0(j))$. $\quad \square$

Since the case J is empty was already addressed in Proposition 4.1, we suppose that $J=\lbrace 1,\ldots ,N\rbrace$ for some $N \in \mathbb {Z}_+$. The space of $\mathcal {L}_1,\mathcal {L}_{\mathfrak {s}}$ operators can be identified with the linear space of free coefficients in their defining formulas (6.18), (6.19). Because of the above Lemma, we have a (nonlinear) map between the two spaces, which satisfies the constraints $\sum _{i=1}^n \bar{r}^{d_i}=\Phi _0(r)$ and $\sum _i\frac{s^{d_i}_0(j)}{w_j^{d_i}}=\Phi _0(X^0(j))$, $j=1,\ldots ,N$.

Definition 6.22

Let

$$\begin{aligned} U_N=\bigoplus _{j=1}^N\bigoplus _{i=1}^{h^{\vee }-1}\{ X^i(j) \in \mathfrak {g}^i \},\qquad V_N=\bigoplus _{j=1}^N\bigoplus _{i=1}^{h^{\vee }-1}V_i(j), \end{aligned}$$

where $V_i(j)=\text{ span } \lbrace s^{d_l}_i(j) \in \mathfrak {s}^{d_l}, i\le d_l\le h^{\vee }-1 \rbrace$. For every $(r,X^0(1),\ldots ,X^0(N)) \in \mathfrak {h}^{\oplus N+1}$, we denote

$$\begin{aligned} F=F^{r,X^0(1),\ldots ,X^0(N)}:U_N \rightarrow V_N, \end{aligned}$$

(6.20)

the map which associates to an operator (6.19) its canonical form.

Recall from the local theory that, fixed the part of total degree $-1$ by the choice of an element $X^0 \in \mathfrak {h}$, there is a map $\Phi ^{X^0}$ from the space of 1-Bessel operators to the space of the corresponding $\mathfrak {s}$-Bessel operators. In the following theorem we prove that the map (6.20) is bijective if and only if, for every $j=1,\ldots ,N$, $X^0(j)$ is such that the map $\Phi ^{X^0(j)}$ is bijective. Due to Proposition 6.17, the latter conditions are verified in an open and dense subset of the parameters $X^0(j) \in \mathfrak {h}^{\oplus N}$. In particular they are verified, by Corollary 6.18, when $X^0(j)=-\theta ^\vee$ for all $j=1,\ldots ,N$, which is the case relevant for the Quantum $\widehat{\mathfrak {g}}$-KdV opers.

Theorem 6.23

The map $F^{r,X^0(1),\ldots ,X^0(N)}$ is bijective if and only if for every $j=1,\ldots ,N$ the map $\Phi ^{X^0(j)}$ is bijective.

Proof

Fix $m \in \lbrace 1,\ldots ,N\rbrace$, and consider the localisation at $w_m$ of the operators $\mathcal {L}_1$ of type (6.18) and $\mathcal {L}_{\mathfrak {s}}$ of type (6.19). We have

$$\begin{aligned}&(\mathcal {L}_1)_{w_m}=\partial _x+f+\sum _{l,k\ge 0}\frac{u^l_k}{x^{1-k}}, \qquad u^l_k \in \mathfrak {g}^l \end{aligned}$$

(6.21)

$$\begin{aligned}&(\mathcal {L}_{\mathfrak {s}})_{w_m}=\partial _x+f+\sum _{l,k\ge 0}\frac{t^{d_l}_k}{x^{d_l+1-k}},\qquad t^{d_l}_k \in \mathfrak {s}^{d_l}. \end{aligned}$$

(6.22)

The coefficients $u^l_k$ appearing in (6.21) can be written in terms of the original variables of $\mathcal {L}_1$ as

$$\begin{aligned} u^i_{0}=\frac{X^i(m)}{w_m^i}, \qquad u^{l}_{l-i}=\sum _{j \in J {\setminus } \lbrace m\rbrace }a_{i,l,j}X^{l}(j), \quad l=0,\ldots i-1. \end{aligned}$$

(6.23)

for some complex coefficients $a_{i,l,k}$. We define $\widetilde{V_i}(m)$, $i=1,\ldots , h^{\vee }-2$, as the subspace of coefficients $t^{d_l}_k$ appearing in (6.22) which have total degree $i-1$ and principal degree at least i. Namely, we have

$$\begin{aligned} \widetilde{V_i}(m)=\lbrace t^{d_l}_i, d_l\ge i \rbrace . \end{aligned}$$

For each pair of indices m, i, the Gauge transformation $\mathcal {L}_1\rightarrow \mathcal {L}_{\mathfrak {s}}=Y.\mathcal {L}_1$ from $\mathcal {L}_1$ to its canonical form $\mathcal {L}_{\mathfrak {s}}$ induces a map $\overline{F_i}(m): U_N \rightarrow \widetilde{V_i}(m)$, obtained by first localizing the image operator $Y.\mathcal {L}_1$ at $z=w_m$ and then restricting to the terms in $\widetilde{V_i}(m)$. As we prove below, the map $\overline{F_i}(m)$ admits the decomposition

$$\begin{aligned} \overline{F_i}(m)=A_i^{X^0(m)}(\frac{X^i(m)}{w_m^i})+\overline{P}_{i,m}\;, \end{aligned}$$

(6.24)

where for each $X^0\in \mathfrak {h}$ the map $A_i^{X^0}: \mathfrak {g}^i \rightarrow \widetilde{V}_i$, is linear and coincides with the map $A_i$ defined in Lemma 6.15, while $\bar{P}_{i,m}$ is a function of the variables $X^{l}(m)$, with $l\le i-1$, and $m \in \lbrace 1,\ldots , N \rbrace$.

In order to prove the decomposition (6.24), we adapt the proof of Lemma 6.15 to the present case. Let $(Y)_{w_m}$ be the localization at $z=w_m$ of the Gauge transform Y mapping $\mathcal {L}_1$ to $\mathcal {L}_{\mathfrak {s}}$. We obtain $(Y)_{w_m}\in \mathcal {N}_{loc}^{\ge 0}$ as the direct limit $\varinjlim Y_i$, where $Y_i$ maps a truncation of $(\mathcal {L}_1)_{w_m}$ to its canonical form, up to terms of high (enough) total and principal degrees in such a way that the functions $\overline{F}_i(m)$, with $i=1,\ldots , h^{\vee }-1$, are completely determined by the action of $Y_0,\ldots ,Y_{h^{\vee }-2}$ only. This is done as follows. Let $\mathcal {L}_i$ be the projection of $(\mathcal {L}_1)_{w_m}$ onto the subspace of total degree $\le i-1$ and principal degree $\le i$. Then from (6.21) we get

$$\begin{aligned} \mathcal {L}_i=\sum _{l=0}^{i}\sum _{k=0}^{l+i}\frac{u^l_m}{x^{1-k}}. \end{aligned}$$

Then we look for $Y_i$ such that $\Pi _f(Y_{i} \mathcal {L}_{i})$ is a linear combination of terms of total degree $>i-1$, and moreover the terms of total degree equal to i have principal degree $\le i$.

For $i=0$ we choose the Gauge transformation $Y_0=x^{-\rho ^\vee }Y$, with $Y \in \mathcal {N}$, which maps $\partial _x+f+x^{-1}u^0_0$ to $\partial _x+f+\sum _{l=1}^n\frac{t^{d_l}_0}{x^{d_l+1}}$. We then reconstruct $Y_i$ recursively as follows. (1) We look for $y_{i}=\sum _{k\ge 1}\frac{y^{i+k}}{x^{k}}$, with $y^{i+k}\in \mathfrak {g}^{i+k}$ such that the projection

$$\begin{aligned} \Pi _f(\exp {y_{i}}Y_{i-1}. \mathcal {L}_{i}) \end{aligned}$$

only contains terms of total degree greater than i. Notice that $y_i$ is non-trivial only if $i\le h^{\vee }-2$. (2) We obtain $Y_{i}$ as $\exp {y'_{i}}\exp {y_{i}}Y_{i-1}$ where $y'_{i}=\sum _{k=0}^{l}\frac{p^{k}}{x^{k-i-1}}$, for some $p^k \in \mathfrak {g}^k$.

We implement (1) following the proof of Lemma 6.15, and we obtain a linear system for $y^{i+k}$ and thus for $\overline{F}_i(m)$. This coincides with system (6.16) after we rename the variables $X^i \rightarrow y^i_0=\frac{X^i_0(m)}{w^m}$, $\Phi _i \rightarrow \overline{F}_i(m)$. In this system, the known terms b’s are shown, recursively, to depend on the coefficients $u^l_k, l\le i-1,k\le l$ of the local expansion of $\mathcal {L}_1$; after (6.23)), these terms depend on $X^l(j)$, for $l\le i-1$, and $j =1,\ldots ,N$. Hence the same reasoning as in  6.15 after equation (6.16) proves the decomposition (6.24).

We now prove that the decomposition (6.24) implies the thesis. First we compute the coefficients $t^{d_l}_i$ spanning $\widetilde{V}_i(m)$ in terms of the original coeffcients of the oper $\mathcal {L}_{\mathfrak {s}}$. We have

$$\begin{aligned} t^{d_l}_i=\frac{s^{d_l}_i(m)}{w_m^{d_l}}+\sum _{0\le k \le i-1}b_{i,l,k}s^{d_l}_{k}(m), \end{aligned}$$

(6.25)

for some complex $b_{i,l,k}$. By above formulas, we have an invertible map

$$\begin{aligned} \bigoplus _{m=1}^N \bigoplus _{i\ge 1} V_i(m) \rightarrow \bigoplus _{m=1}^N\bigoplus _{i\ge 1}\widetilde{V_i}(m), \end{aligned}$$

which associates to an oper of type (6.18) the totaliy of local coefficients belonging to the spaces $\widetilde{V_i}(m)$’s. It follows that $F:U_N \rightarrow V_N$ is bijective if and only if the map

$$\begin{aligned} \overline{F}=\sum _{m=1}^N \sum _{i=1}^{h^{\vee }-1} \overline{F}_i(m):U_N \rightarrow \bigoplus _{m=1}^N \bigoplus _{i=1}^{h^{\vee }-1} \widetilde{V_i}(m) \end{aligned}$$

is bijective. Due to the decomposition (6.24), one deduces recursively that $\overline{F}$ is bijective if and only if the maps $\overline{F}_j(m)$ are bijective for every i (and m), if and only if the linear maps $A_i^{X^0(m)}:\mathfrak {g}^i \rightarrow \widetilde{V}_i(m)$ are bijective for every m and i. After Lemma 6.15(ii), the linear maps $A_i^{X^0(m)}:\mathfrak {g}^i \rightarrow \widetilde{V}_i(m)$ are bijective for every $m=1,\ldots ,N$ and every $i=1\ldots h^{\vee }-1$ if and only if the maps $\Phi ^{X^0(m)}$ are bijective for all $m=1\ldots N$. $\quad \square$

Theorem 6.1 is a direct corollary of the previous theorem:

Proof of Theorem 6.1

The case when J is empty was proved in Lemma 4.1. Here we suppose $J=\lbrace 1\ldots N \rbrace$ with $N \in \mathbb {Z}_+$. Using the map $\Phi _0$ introduced above, then we can restate Proposition 4.7 as follows: an oper $\mathcal {L}$ of the form (4.8) and satisfying the Assumptions 1, 2, 3 can be written as

$$\begin{aligned} \mathcal {L}=\mathcal {L}_{\mathfrak {s}}+z^{-h^{\vee }+1} (1+ \lambda z^{-\hat{k}})e_{\theta }, \end{aligned}$$

where $\mathcal {L}_{\mathfrak {s}}$ is the oper (6.18) and

$$\begin{aligned} \Phi _0(r)&=\, \bar{r},\\ \Phi _0(-\theta ^\vee )&=\sum _i^n \frac{s^{d_i}_0(j)}{w_j^{d_i}}, \qquad j=1,\ldots ,N. \end{aligned}$$

Due to Corollary 6.18, the map $\Phi ^{-\theta ^\vee }$ is bijective. Therefore, Theorem 6.23 implies that the operator $\mathcal {L}_{\mathfrak {s}}$ is Gauge equivalent to a unique operator of the form (6.19), with $X^0(j)=-\theta ^\vee$ for $j=1,\ldots ,N$. Since the term $z^{-h^{\vee }+1} (1+ \lambda z^{-\hat{k}})e_{\theta }$ is Gauge invariant under $\mathcal {N}(K_{\mathbb {P}^1})$, the thesis follows. $\quad \square$

8 The Gradation Induced by the Highest Root

In order to proceed with our program, we have to impose on the operators (6.1) the trivial monodromy conditions at any additional singularity $w_j,j \in J$. These operators have locally, at any $w_j,j \in J$, the expansion

$$\begin{aligned} \partial _x+\frac{-\theta ^\vee + \eta }{x}+O(1), \qquad \eta \in \mathfrak {n}^+, \end{aligned}$$

where $\theta ^\vee$ is the dual to the highest root of the Lie algebra, $\theta ^\vee =\nu ^{-1}(\theta )$. As we will see in the next section, for an operator with a simple pole, the monodromy is computed by decomposing its coefficients in the eigenspaces of the adjoint action of the residue$-\theta ^\vee + \eta$. As a necessary preliminary tool, we therefore devote this section to the study of the eigen-decomposition of $\mathfrak {g}$ with respect to the adjoint action of $-\theta ^\vee +\eta$ with $\eta \in \mathfrak {n}^+$.

8.1 The gradation induced by $\theta$

We need the following lemma, which can be found in [30, Section 9].

Lemma 7.1

Let $\alpha ,\beta$ be nonproportional roots. Then

• (i) If $(\alpha | \beta )>0$ then $\alpha -\beta$ is a root. If $(\alpha | \beta )<0$ then $\alpha +\beta$ is a root.

• (ii) Since $\mathfrak {g}$ is simply-laced then $(\alpha | \beta )\in \{-1,0,1\}$.

Let $\theta \in \Delta$ be the highest root of $\mathfrak {g}$, and denote $\theta ^\vee =\nu ^{-1}(\theta )\in \mathfrak {h}$. We want to study the spectrum of ${{\,\mathrm{ad}\,}}\theta ^\vee$ in the adjoint representation. It is clear that for every $\alpha \in \Delta$, if $x\in \mathfrak {g}$ belongs to the root space of $\alpha$ then we have

$$\begin{aligned}{}[\theta ^\vee ,x]=(\theta | \alpha ) x. \end{aligned}$$

Now we can apply Lemma 7.1 (ii) to the case when one of the two roots is $\theta$, the highest root. The only roots proportional to $\theta$ are $\pm \theta$, and we have $(\theta | \pm \theta )=\pm 2$. Due to the lemma, then $(\theta | \beta )\in \{-1,0,1\}$ for every $\beta \in \Delta {\setminus }\{-\theta ,\theta \}$. The spectrum of ${{\,\mathrm{ad}\,}}\theta ^\vee$ in the adjoint representation is then given by

$$\begin{aligned} \sigma (\theta ^\vee )= {\left\{ \begin{array}{ll} \left\{ -2,-1,0,1,2\right\} \qquad &{} \quad \text { if } \mathfrak {g}\ne \mathfrak {sl}_2\\ \left\{ -2,0,2\right\} \qquad &{} \quad \text { if } \mathfrak {g}= \mathfrak {sl}_2, \end{array}\right. } \end{aligned}$$

(7.1)

and we obtain a $\mathbb {Z}$-gradation of $\mathfrak {g}$ as

$$\begin{aligned} \mathfrak {g}=\bigoplus _{i=-2}^2\mathfrak {g}_i, \qquad \mathfrak {g}_i=\left\{ x\in \mathfrak {g}\;\vert \; [\theta ^\vee ,x]=ix\right\} , \end{aligned}$$

(7.2)

which we call the highest root gradation. We denote

$$\begin{aligned} \pi _j: \;\mathfrak {g}\rightarrow \mathfrak {g}_j \end{aligned}$$

(7.3)

the natural projection from $\mathfrak {g}$ onto the j-th component of the gradation, and we set

$$\begin{aligned} x_j=\pi _j(x),\qquad x\in \mathfrak {g}. \end{aligned}$$

(7.4)

We describe in more detail the structure of the gradation (7.2). Note that $\mathfrak {h}\subset \mathfrak {g}_0$, and that $\mathfrak {n}_+$ uniquely decomposes as

$$\begin{aligned} \mathfrak {n}_+=(\mathfrak {g}_0\cap \mathfrak {n}_+)\oplus \mathfrak {g}_1\oplus \mathfrak {g}_2. \end{aligned}$$

(7.5)

Let $I_{\theta ^\vee }=\left\{ j\in I\;\vert \;\langle \theta ^\vee ,\alpha _j\rangle =0\right\} \subset I,$ and denote by $\mathfrak {g}[I_{\theta ^\vee }]$ the semisimple Lie algebra generated by $\{e_{i}, f_{i}, i\in I_{\theta ^\vee }\}$. Then, we have

$$\begin{aligned} \mathfrak {g}_0=\mathfrak {g}[I_{\theta ^\vee }]\oplus \bigoplus _{i\in I{\setminus } I_{\theta ^\vee }} \mathbb {C}\alpha _i^\vee . \end{aligned}$$

(7.6)

Remark 7.2

The set $I_{\theta ^\vee }$ is depicted in Table 1 as the subset of white vertices in the Dynkin diagram. The subalgebra $\mathfrak {g}[I_{\theta ^\vee }]$ is isomorphic to the semi-simple Lie algebra whose Dynkin diagram is obtained by the Dynkin diagram of $\mathfrak {g}$, by removing the black vertices and all the edges to which they are connected. These subalgebras are explicitly computed in Table 2. Moreover, setting $\mathfrak {p}=\mathfrak {g}_0\oplus \mathfrak {u}$, with $\mathfrak {u}=\bigoplus _{i> 0}\mathfrak {g}_i,$ then $\mathfrak {p}$ is a parabolic subalgebra of $\mathfrak {g}$, with $\mathfrak {g}_0$ a reductive (Levi) subalgebra and $\mathfrak {u}$ the nilradical of $\mathfrak {p}$.

Table 1 Dynkin diagrams for simple Lie algebras of ADE type

Table 2 The (dual) Coxeter number $h^\vee$, the semi-simple subalgebra $\mathfrak {g}[I_{\theta }]$, its dimension, and the set $I{\setminus } I_\theta$ for any simply-laced Lie algebra $\mathfrak {g}$

The dimension of the graded components of (7.2) is computed in the following

Proposition 7.3

Let $\mathfrak {g}$ be simply laced, and consider the gradation (7.2). Then a) $\dim \mathfrak {g}_i=\dim \mathfrak {g}_{-i}$, $i=0,1,2$. b) We have:

$$\begin{aligned} \dim \mathfrak {g}_0&=n(h^\vee +1)-4h^\vee +6,\\ \dim \mathfrak {g}_1&=2(h^\vee -2), \\ \dim \mathfrak {g}_{2}&=1. \end{aligned}$$

In particular, $\mathfrak {g}_{2}=\mathfrak {g}^{h^\vee -1}$ coincides with the root space of the highest root $\theta$.

Proof

Part a) is obvious. For part b) we proceed as follows. The dimension of $\mathfrak {g}_2$ is computed by first noticing that the roots proportional to $\theta$ are $\pm \theta$, with $(\theta ,\pm \theta )=\pm 2$, and then using Lemma 7.1 (ii) which implies that $\mathfrak {g}_2=\mathfrak {g}^{h^\vee -1}$, the root space of the highest root $\theta$. In particular, $\dim \mathfrak {g}_2=1$. (Incidentally, the same argument shows that $\mathfrak {g}_{-2}=\mathfrak {g}^{1-h^\vee }$, the root space of the lowest root $-\theta$, so that $\dim \mathfrak {g}_{-2}=1$). To compute $\dim \mathfrak {g}_0$ note that due to (7.6) we have $\dim \mathfrak {g}_0=\dim \mathfrak {g}[I_{\theta ^\vee }]+\#(I{\setminus } I_{\theta })$. By looking in Table 2 at the values of $h^\vee$, $\dim \mathfrak {g}[I_{\theta ^\vee }]$ and $\#(I{\setminus } I_{\theta })$, one proves (case by case) that $\dim \mathfrak {g}_0=n(h^\vee +1)-4h^\vee +6$. Finally, using part a) we get $\dim \mathfrak {g}=\dim \mathfrak {g}_0+2\dim \mathfrak {g}_1+2\dim \mathfrak {g}_2$. Substituting the values of $\dim \mathfrak {g}_0$ and $\dim \mathfrak {g}_2$ just obtained, and recalling that dimension of the simple Lie algebra $\mathfrak {g}$ is given by $n(h+1)$, where n is the rank and h the Coxeter number (with $h=h^\vee$ since $\mathfrak {g}$ is simply laced), then the last identity becomes $n(h^\vee +1)=n(h^\vee +1)-4h^\vee +6+2\dim \mathfrak {g}_1+2$, which gives $\dim \mathfrak {g}_1=2(h^\vee -2)$. $\quad \square$

8.2 The gradation induced by $R=-\theta ^\vee +\eta$

Later we will be interested in gradations of $\mathfrak {g}$ induced by elements of the form $R=-\theta ^\vee +\eta$, with $\eta \in \mathfrak {n}_+$. We begin by recalling the definition of Jordan–Chevalley decomposition.

Definition 7.4

Let $R \in \mathfrak {g}$. There exists a unique decomposition, named Jordan–Chevalley decomposition, of the following form $R=R_s+R_n$, with $R_s$ semisimple, $R_n$ nilpotent, and $[R_s,R_n]=0$. We denote $\sigma (R)=\sigma (R_s)$ the spectrum of R in the adjoint representation.

The following lemma will be very useful.

Lemma 7.5

Let $R=-\theta ^\vee +\eta$, with $\eta \in \mathfrak {n}_+$, and write $\eta =\eta _0+\eta _1+\eta _2$ with $\eta _i \in \mathfrak {g}_i$. Then

• 1. $\sigma (R)=\sigma (\theta ^\vee )$;

• 2. R is semisimple if and only if $\eta _0=0$;

• 3. If R is semisimple then

  $$\begin{aligned} R=-e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}\theta ^\vee . \end{aligned}$$

  (7.7)

Proof

Let $\bar{\eta }_1\in \mathfrak {g}$ satisfy $({{\,\mathrm{ad}\,}}\eta _0-1)\bar{\eta }_1=\eta _1$, and $\bar{\eta }_2\in \mathfrak {g}$ be such that $({{\,\mathrm{ad}\,}}\eta _0-2)\bar{\eta }_2=\eta _2+\frac{1}{2}[\bar{\eta }_1,\eta _1]$. Then $\bar{\eta }_i\in \mathfrak {g_i}\subset \mathfrak {n}_+$, $i=1,2$, and $[\bar{\eta }_1,\bar{\eta }_2]=0$. Moreover, we have

$$\begin{aligned} e^{{{\,\mathrm{ad}\,}}(\bar{\eta }_1+\bar{\eta }_2)}R=-\theta ^\vee +\eta _0. \end{aligned}$$

Since $\theta ^\vee$ is semisimple, $\eta _0$ is nilpotent and $[\theta ^\vee ,\eta _0]=0$, then we obtain that

$$\begin{aligned} R_s=-e^{-{{\,\mathrm{ad}\,}}(\bar{\eta }_1+\bar{\eta }_2)}\theta ^\vee ,\qquad R_n=e^{-{{\,\mathrm{ad}\,}}(\bar{\eta }_1+\bar{\eta }_2)}\eta _0 \end{aligned}$$

(7.8)

are, respectively, the semisimple and nilpotent parts of the Jordan–Chevalley decomposition of R. From this, we obtain: (1) $\sigma (R)=\sigma (\theta ^\vee )$, (2) R is semisimple if and only if $\eta _0=0$, (3) if R is semisimple so that $\eta _0=0$, we have $\bar{\eta }_1=-\eta _1$ and $\bar{\eta }_2=-\frac{\eta _2}{2}$, so that $R_s$ given by (7.8) coincides with R given by (7.7). $\quad \square$

Let us now consider the gradation

$$\begin{aligned} \mathfrak {g}=\bigoplus _{i\in \sigma (R)}\mathfrak {g}_i(R), \qquad \mathfrak {g}_i(R)=\left\{ x\in \mathfrak {g}\;\vert \; [R,x]=ix\right\} , \end{aligned}$$

(7.9)

in case R is semi-simple. Note that due to (7.7), the gradation (7.9) is conjugated to the highest root gradation (7.2), namely

$$\begin{aligned} \mathfrak {g}_i(R)=\left\{ e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}x\,\vert \,x\in \mathfrak {g}_{-i}\right\} . \end{aligned}$$

(7.10)

For $j\in \sigma (R)$ we denote by $\pi ^{R}_j$ the natural projection from $\mathfrak {g}$ onto $\mathfrak {g}_j(R)$. Note that from (7.10) we have that

$$\begin{aligned} \pi _j^R=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)} \pi _{-j} e^{-{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}. \end{aligned}$$

We write this formula in a very explicit form that we will need in the following section, when dealing with trivial monodromy conditions. Let $x\in \mathfrak {g}$ and denote by $x_i=\pi _i(x)$ the projection (7.3) of x onto $\mathfrak {g}_i$, then we have

$$\begin{aligned} \pi ^R_{2}(x)&=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}x_{-2}\nonumber \\ \pi ^R_{1}(x)&=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}(x_{-1}-{{\,\mathrm{ad}\,}}_{\eta _1}x_{-2})\nonumber \\ \pi ^R_{0}(x)&=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}(x_0-{{\,\mathrm{ad}\,}}_{\eta _1} x_{-1}+ \frac{1}{2}{{\,\mathrm{ad}\,}}^2_{\eta _1}x_{-2}-\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _2}x_{-2})\nonumber \\ \pi ^R_{-1}(x)&=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}(x_1-{{\,\mathrm{ad}\,}}_{\eta _1}x_0+\frac{1}{2}{{\,\mathrm{ad}\,}}^2_{\eta _1}x_{-1}- \frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _2}x_{-1}-\frac{1}{6}{{\,\mathrm{ad}\,}}^3_{\eta _1}x_{-2}\nonumber \\&\quad +\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _1}{{\,\mathrm{ad}\,}}_{\eta _2}x_{-2})\nonumber \\ \pi ^R_{-2}(x)&=x_2-{{\,\mathrm{ad}\,}}_{\eta _1}x_1-\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _2}x_{0}+ \frac{1}{2}{{\,\mathrm{ad}\,}}^2_{\eta _1}x_0-\frac{1}{6}{{\,\mathrm{ad}\,}}^3_{\eta _1}x_{-1}+\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _1}{{\,\mathrm{ad}\,}}_{\eta _2}x_{-1}\nonumber \\&\quad +\frac{1}{24}{{\,\mathrm{ad}\,}}^4_{\eta _1}x_{-2}+\frac{1}{8}{{\,\mathrm{ad}\,}}^2_{\eta _2}x_{-2}-\frac{1}{2}{{\,\mathrm{ad}\,}}^2_{\eta _1}{{\,\mathrm{ad}\,}}_{\eta _2}x_{-2}. \end{aligned}$$

(7.11)

The above identities have been obtained by means of the following expansion:

$$\begin{aligned} e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}&=1+{{\,\mathrm{ad}\,}}\eta _1+\frac{1}{2}{{\,\mathrm{ad}\,}}\eta _2+\frac{1}{2}{{\,\mathrm{ad}\,}}^2\eta _1+ \frac{1}{2}{{\,\mathrm{ad}\,}}\eta _1{{\,\mathrm{ad}\,}}\eta _2+\frac{1}{8}{{\,\mathrm{ad}\,}}^2\eta _2\nonumber \\&\quad +\frac{1}{6}{{\,\mathrm{ad}\,}}^3\eta _1+\frac{1}{4}{{\,\mathrm{ad}\,}}^2\eta _1{{\,\mathrm{ad}\,}}\eta _2+\frac{1}{24}{{\,\mathrm{ad}\,}}^4\eta _1. \end{aligned}$$

(7.12)

Remark 7.6

Note that $\mathfrak {g}_1(R)\oplus \mathfrak {g}_2(R)\subset \mathfrak {n}_+$, while $\mathfrak {g}_0(R)=\mathfrak {h}^{R}\oplus (\mathfrak {g}_0(R)\cap \mathfrak {n}_+)$, where $\mathfrak {h}^{R}$ is the Cartan subalgebra conjugated to $\mathfrak {h}$ under the automorphism $\exp ({{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2))$.

8.3 A symplectic subspace

Consider the vector subspace

$$\begin{aligned} \mathfrak {t}=\mathbb {C}\theta ^\vee \oplus \mathfrak {g}_1\oplus \mathfrak {g}_2\subset \mathfrak {g}, \end{aligned}$$

(7.13)

which due to Proposition 7.3 is even dimensional, of dimension $\dim \mathfrak {t}=2(h^\vee -1)$. Note that we can write $\mathfrak {t}$ as $\mathfrak {t}=\mathbb {C}\theta ^\vee \oplus [\theta ^\vee ,\mathfrak {n}_+]$. Define on $\mathfrak {t}$ the skew-symmetric bilinear form

$$\begin{aligned} \omega (x,y)=(E_{-\theta }\vert [x,y]), \qquad x,y\in \mathfrak {t}, \end{aligned}$$

(7.14)

where $(\cdot \vert \cdot )$ is the normalized invariant bilinear form (1.6) on $\mathfrak {g}$, and $E_{-\theta }$ is the lowest weight vector of $\mathfrak {g}$ introduced in Sect. 1. The form (7.14) is also non-degenerate, therefore it defines a symplectic structure on $\mathfrak {t}$. We outline here two different approaches to prove this fact. The first approach is based on the fact [37] that $\mathfrak {t}=({{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}E_{-\theta })^\perp$, the orthogonal complement (with respect to $(\cdot \vert \cdot )$ ) of the vector subspace ${{\,\mathrm{Ker}\,}}{{\,\mathrm{ad}\,}}E_{-\theta }$. Therefore, $\mathfrak {t}$ is a symplectic leaf of the ‘frozen’  Lie–Poisson structure [39], and (7.14) is nothing but the induced symplectic form on $\mathfrak {t}$.

For the second approach we present a canonical basis for (7.14), according to the following construction. Recall the root vectors $E_\alpha$, $\alpha \in \Delta$, of $\mathfrak {g}$ introduced in Sect. 1, satisfying the commutation relations (1.13). From (7.2) we have that $E_{\alpha }\in \mathfrak {g}_i$ if and only if $(\alpha | \theta )=i$, so that $\mathfrak {g}_1=\langle E_\alpha , (\theta |\alpha )=1\rangle$ and $\mathfrak {g}_2=\langle E_\theta \rangle$. In order to deal with elements in $\mathfrak {t}$, we define the set $\Theta \subset \mathfrak {h}^*$ as

$$\begin{aligned} \Theta =\{0,\theta \}\cup \{\alpha \in \Delta \,|\, (\alpha | \theta )=1\}. \end{aligned}$$

(7.15)

Denoting $E_0=\theta ^\vee /2$ then $\{E_\alpha \,|\, \alpha \in \Theta \}$ is a basis for $\mathfrak {t}$. Recall the bimultiplicative function $\varepsilon _{\alpha ,\beta }$ introduced in Sect. 1.

Lemma 7.7

If $\alpha \in \Theta$, then $\theta -\alpha \in \Theta$ and $\theta -\alpha \ne \alpha$. Moreover, $\varepsilon _{\alpha ,\theta }=-\varepsilon _{\theta -\alpha ,\theta }$.

Proof

If $\alpha$ is equal to $\theta$ or 0 the only nontrivial assertion is the last, where due to Lemma 1.1 we have $\varepsilon _{0,\theta }=1$ and $\varepsilon _{\theta ,\theta }=-1$. Now let $\alpha \in \Theta$ with $(\alpha | \theta )=1$. Then, due to Lemma 7.1(i) we have $\theta -\alpha \in \Delta$, and $(\theta -\alpha | \theta )=1$, so that $\theta -\alpha \in \Theta$. Moreover, $\theta -\alpha =\alpha$ implies $\theta =2\alpha$ for $\alpha \in \Delta$, which is impossible. Thus, $\theta -\alpha \ne \alpha$. Finally, due to Lemma 1.1, we have $\varepsilon _{\theta -\alpha ,\theta }=\varepsilon _{\theta ,\theta }\varepsilon _{-\alpha ,\theta }=-\varepsilon _{\alpha ,\theta }$. $\quad \square$

The function $\varepsilon$ takes values $\pm 1$. We introduce the subset

$$\begin{aligned} \widetilde{\Theta }=\{\alpha \in \Theta \,|\,\varepsilon _{\theta ,\alpha }=1\}\subset \Theta , \end{aligned}$$

(7.16)

and due to Lemma 7.7 we have $\Theta =\{\alpha ,\theta -\alpha \, |\, \alpha \in \widetilde{\Theta }\}$.

Proposition 7.8

For $x,y\in \mathfrak {t}$, with $x=\sum _{\alpha \in \Theta }x^\alpha E_\alpha$ and $y=\sum _{\alpha \in \Theta }y^\alpha E_\alpha$ then (7.14) takes the form

$$\begin{aligned} \omega (x,y)=-\sum _{\alpha \in \widetilde{\Theta }}\left( x^{\alpha }y^{\theta -\alpha }-x^{\theta -\alpha } y^{\alpha }\right) , \end{aligned}$$

where $\widetilde{\Theta }$ is given by (7.16). Thus, $\{E_\alpha \,|\,\alpha \in \Theta \}$ is a canonical basis for $\mathfrak {t}$.

Proof

Let c be the coefficient of $E_\theta$ in the commutator [X, Y]. Due to (7.14) and (1.15) we have $\omega (X,Y)=-c$. An explicit computation gives

$$\begin{aligned} c=\sum _{\alpha \in \Theta }\epsilon _{\theta ,\alpha }x^\alpha y^{\theta -\alpha }=\sum _{\alpha \in \widetilde{\Theta }}(x^\alpha y^{\theta -\alpha }-x^{\theta -\alpha }y^\alpha ), \end{aligned}$$

where in the last step we used Lemma 7.7. $\quad \square$

The symplectic space $\mathfrak {t}$ will play an important role in Sect. 9, when computing the trivial monodromy conditions for quantum-KdV opers.

Remark 7.9

In order to obtain a canonical basis for the form (7.14) we chose in the construction above $E_0=\theta ^\vee /2$. In the computations we will perform in Sects. 9 and 10 it will be slightly more convenient to choose $E_0=\theta ^\vee$. We will point out our choice whenever required.

9 Trivial Monodromy at a Regular Singular Point

In this Section—following [2]—we consider an arbitrary linear operator with a first order pole, and we derive necessary and sufficient conditions (on its Laurent series) to have trivial monodromy. We then specialize to the case of the localisation of a Quantum $\widehat{\mathfrak {g}}$-KdV oper (6.1) at an additional singularity.

Let $x=0$ be the singular point, and assume that the operator has the expansion

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{R}{x}+\sum _{k\ge 0} a^k x^k, \end{aligned}$$

(8.1)

with $R,a^k \in \mathfrak {g}$.

Definition 8.1

We say that the operator (8.1) has trivial monodromy at $x=0$ if, for any finite dimensional $\mathfrak {g}$-module V, the differential equation $\mathcal {L}\psi =0$, with $\psi : \mathbb {C} \rightarrow V$, has trivial monodromy.

It is well known that the eigenvalues of the monodromy matrix are of the form $\exp {2 i \pi \lambda }$, where $\lambda$ an eigenvalue of R. Since we look for conditions on $\mathcal {L}$ such that the monodromy matrix is the identity in every representation, we must restrict to the case where R has integer eigenvalues in any finite-dimensional representation of $\mathfrak {g}$. For this reason, we assume that the semi-simple part $R_s$ of R in the Jordan–Chevalley decomposition is conjugated to an element of the co-root lattice $Q^\vee$ of $\mathfrak {g}$. In fact, see Proposition 8.3(5), the operator (8.1) has trivial monodromy if and only if it has trivial monodromy in the adjoint representation and $R_s$ is conjugated to an element of the co-root lattice.

Because of our assumption on $R_s$, we have the $\mathbb {Z}$-gradation

$$\begin{aligned} \mathfrak {g}=\bigoplus _{i\in \sigma (R)}\mathfrak {g}_i(R), \qquad \mathfrak {g}_i(R)=\left\{ x\in \mathfrak {g}\;\vert \; [R_s,x]=ix\right\} . \end{aligned}$$

(8.2)

In order to compute the monodromy of $\mathcal {L}$ at $x=0$ we transform it into its aligned form. The following definition is adapted from [2].

Definition 8.2

[2] The operator (8.1) is said to be aligned (at $x=0$) if $a^i\in \mathfrak {g}_{-i-1}(R)$ for $i\ge 0$.

Proposition 8.3

[2] Let $\mathcal {L}$ be a connection with local expansion (8.1), such that $R_s$ is conjugated to an element of the co-root lattice $Q^\vee$, and let $m=\max \sigma (R)$. Let moreover $G[[x]]_1$ be the sub-group of Gauge transformations of the form $G[[x]]_1=\left\{ 1+\sum _{i\ge 1} M_i x^i\;\vert \; M_i\in \mathfrak {g}\right\}$ and positive radius of convergence.

• 1. $\mathcal {L}$ is equivalent to an aligned connection by a transformation in $G[[x]]_1$.

• 2. The monodromy of $\mathcal {L}$ coincides with the monodromy of the aligned connection equivalent to it.

• 3. If the first m coefficients $a^0,\ldots a^{m-1}$ appearing in the expansion (8.1) of $\mathcal {L}$ are such that $a^i \in \mathfrak {g}_{-i-1}(R)$ for $i=0,\ldots ,m-1$, then $\mathcal {L}$ is conjugated in $G[[x]]_1$ to the aligned connection $\partial _x+x^{-1}R+\sum _{i=0}^{m-1} a^i x^i$.

• 4. The monodromy of the connection $\mathcal {L}$ depends only on the first $m+1$ coefficients of the expansion of $\mathcal {L}$ at $x=0$, namely $R, a^0,\ldots ,a^{m-1}$.

• 5. Let $\mathcal {L}$ be aligned. The monodromy is trivial if and only if $R_n=0$, and $a^k=0$ for $k=0,\ldots ,m-1$.

Proof

The proof can be found in [2, Section 3]. Here we just comment on part (5). Let $\mathcal {L}$ be aligned, namely of the form

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{R}{x}+\sum _{k=0}^{m-1} a^k x^k, \qquad a^k\in \mathfrak {g}_{-k-1}(R), \end{aligned}$$

(8.3)

with $m\le \max \sigma (R)$. Since $R_s$ belongs to the co-root lattice, the Gauge transformation $x^{{{\,\mathrm{ad}\,}}R_s}$ is single valued in any finite dimensional $\mathfrak {g}$-module. We have

$$\begin{aligned} x^{{{\,\mathrm{ad}\,}}R_s} \mathcal {L}=\partial _x+\frac{R_n+\sum _{k=0}^{m-1} a^k}{x}, \end{aligned}$$

(8.4)

and the monodromy of the above operator coincides with the monodromy of $\mathcal {L}$. We now prove that the element $R_n+\sum _{k=0}^m a^k$ is nilpotent. Indeed, by definition $R_n\in \mathfrak {g}_0(R)$, so that ${{\,\mathrm{ad}\,}}_{R_n}:\mathfrak {g}_j(R)\rightarrow \mathfrak {g}_j(R)$, while by hypothesis $a_k\in \mathfrak {g}_{-k-1}(R)$, so that ${{\,\mathrm{ad}\,}}_{a_k}:\mathfrak {g}_j(R)\rightarrow \mathfrak {g}_{j-k-1}(R)$, with $k\ge 0$. Suppose that $\lambda \ne 0$ is a non trivial eigenvalue of $R_n+\sum _{j=0}^m a^j$ with eigenvector $0\ne y=\sum _k y_k$ and $y_k \in \mathfrak {g}_k(R)$. If K is the minimum integer such that $y_k\ne 0$, then $y_K$ is an eigenvector of $R_n$ with non-trivial eigenvalue $\lambda$, which is a contradiction, because $R_n$ is nilpotent.

Now let V be a non-trivial finite dimensional $\mathfrak {g}$-module, let $\{\psi _i,\,i=1,\ldots ,\dim V\}$ be a basis of V. Then the functions $\Psi _i=x^{-\big (R_n+\sum _{j=0}^{m-1} a^j\big )}\psi _i$, with $i=1 \ldots \dim V$, satisfy $(x^{{{\,\mathrm{ad}\,}}R_s}\mathcal {L})\Psi _i=0$ and form a basis of solutions. The monodromy matrix is therefore given by

$$\begin{aligned} \mathcal {M}=\exp \big (-2 i \pi (R_n+\sum _{j=0}^{m-1} a^j)\big ). \end{aligned}$$

Since $R_n+\sum _{j=0}^{m-1} a^j$ is nilpotent then the monodromy is trivial if and only if $R_n+\sum _{j=0}^{m-1} a^j=0$, from which the thesis follows. $\quad \square$

Now let us consider what happens in the particular case of a quantum $\mathfrak {g}$-KdV oper (6.1). Localising at $w_j$, we obtain a connection of the form (8.1) with $R=-\theta ^\vee +\eta ,$ where $\eta =\sum _{i \ge 1} w_j^{-i} X^i(j) \in \mathfrak {n}^+$. Due to (7.5) we can write

$$\begin{aligned} R=-\theta ^\vee +\eta _0+\eta _1+\eta _2, \end{aligned}$$

with $\eta _0\in \mathfrak {g}_0\cap \mathfrak {n}_+$, $\eta _1\in \mathfrak {g}_1$ and $\eta _2\in \mathfrak {g}_2$. We can then apply Lemma 7.5 from which we obtain that $\sigma (R)=\sigma (\theta ^\vee )$. In particular, $\max \sigma (R)=2$, so that according to Proposition 8.3(4), the monodromy of the quantum KdV opers at $z=w_j$ depends only on the first three terms of the Laurent expansion. In the next theorem we derive necessary and sufficient conditions for the trivial monodromy of an operator of the form

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{R}{x}+a+bx+O(x^2), \end{aligned}$$

(8.5)

with $R=-\theta ^\vee +\eta _0+\eta _1+\eta _2$ and $a,b\in \mathfrak {g}$.

Theorem 8.4

The operator (8.5) has trivial monodromy at $x=0$ if and only if

$$\begin{aligned}&\eta _0=0, \end{aligned}$$

(8.6a)

$$\begin{aligned}&\pi ^{R}_{-1}(a)=0, \end{aligned}$$

(8.6b)

$$\begin{aligned}&\pi ^{R}_{-2}(b)=[\pi ^{R}_{-2}(a),\pi ^{R}_{0}(a)], \end{aligned}$$

(8.6c)

where $\pi ^{R}_{i}$ denotes the projection of $\mathfrak {g}$ onto $\mathfrak {g}_i(R)=\lbrace x \in \mathfrak {g}, [-\theta ^\vee +\eta _1+\eta _2,x]= ix \rbrace$.

Proof

Let $\mathcal {L}$ be an operator of the form (8.5). Due to Proposition 8.3 the monodoromy of $\mathcal {L}$ coincides with the monodromy of its aligned form, and the residue R of $\mathcal {L}$ coincides with the residue of the aligned operator. Moreover, again by Proposition 8.3, if an aligned oper has trivial monodromy then $R_n=0$. It follows that if the monodromy of $\mathcal {L}$ is trivial then $R_n=0$. By Lemma 7.5, the element $R=-\theta ^\vee +\eta _0+\eta _1+\eta _2$ is semisimple if and only if $\eta _0=0$. This proves condition (8.6a).

By Proposition 8.3, the operator (8.1) with $R=-\theta ^\vee +\eta _1+\eta _2$ is equivalent to an operator of the form

$$\begin{aligned} \mathcal {L}_1=\partial _x+\frac{-\theta ^\vee +\eta _1+\eta _2}{x}+C^0+C^1x + O(x^2), \qquad C^i\in \mathfrak {g}_{-i-1}(R), \, i=0,1, \end{aligned}$$

(8.7)

by a transformation in $G[[x]]_1$, and the monodromy is trivial if and only if $C^0=C^1=0$. The thesis is proved once we show that $C^0=\pi ^{R}_{-1}(a)$ and $C^1=\pi ^{R}_{-2}(b)-[\pi ^{R}_{-2}(a),\pi ^{R}_{0}(a)]$. In order to obtain $C^0,C^1$ we look for a Gauge transformation of the form $e^{x^2 T'}e^{x T}\in G[[x]]_1$, such that $e^{xT}\mathcal {L}=\partial _x+\frac{R}{x}+C^0+O(x)$, and $e^{x^2 T'}e^{x T}.\mathcal {L}=\partial _x+\frac{R}{x}+C^0+C^1 x+O(x^2)$. We have

$$\begin{aligned} e^{x T}\mathcal {L}=\partial _x+\frac{R}{x}+D^0+D^1 x+\cdots , \end{aligned}$$

(8.8)

where

$$\begin{aligned} D^0&=a-T+[T,R] \end{aligned}$$

(8.9)

$$\begin{aligned} D^1&=b+[T,a]+\frac{1}{2}[T,[T,R]]. \end{aligned}$$

(8.10)

Now we look for a T such that $D^0$ is aligned, namely $D^0=\pi _{-1}^R(D^0)$. Writing $T=\sum _i \pi ^R_i(T)$, with $\pi ^R_i(T)\in \mathfrak {g}_i(R)$, and choosing

$$\begin{aligned} \pi _{-1}^R(T)=0, \qquad \pi _j^R(T)=\frac{\pi ^R_j(a)}{j+1},\quad j\ne -1, \end{aligned}$$

(8.11)

we get $D^0=\pi ^R_{-1}(a)$, which is aligned. Hence

$$\begin{aligned} C^0=D^0=\pi ^R_{-1}(a). \end{aligned}$$

(8.12)

This proves the second condition (8.6b). Inserting now (8.9), (8.12), and (8.6b), into (8.10) we obtain

$$\begin{aligned} D^1=b+\frac{1}{2}[T,a]. \end{aligned}$$

(8.13)

Now we look for a $T' \in \mathfrak {g}$ such that $\exp {x^2T'}\exp {x T}.\mathcal {L}=\partial _x+\frac{R}{x}+C^0+C^1 x+ O(x^2)$. By repeating the same steps as above, one shows that

$$\begin{aligned} C^1=\pi ^R_{-2}(D^1)=\pi ^R_{-2}(b)+\frac{1}{2}\pi ^{R}_{-2}([T,a]). \end{aligned}$$

(8.14)

Since $\pi _{-2}^R[T,a]=\sum _{j\ge 0}[\pi _{-j}^R(T),\pi _{-2+j}(a)]$, and using (8.11) together with (8.6b), we obtain that

$$\begin{aligned} C^1=\pi ^R_{-2}(b)-[\pi ^R_{-2}(a),\pi ^R_0(a)], \end{aligned}$$

from which the third condition (8.6c) follows. $\quad \square$

Remark 8.5

The Quantum KdV opers (6.1) depend on two set of unknowns, the location of the poles $w_j,j\in J$ and the local coefficients $X^i(j) \in \mathfrak {g}^i, j \in J$. If $J=\lbrace 1\ldots N\rbrace$, these are $(1+\dim \mathfrak {n}_+) N=(1+\frac{n h^{\vee }}{2} )N$ variables. However, due to the previous theorem, the condition $\pi _0(X^i(j))=0$, holds for every $i=1,\ldots ,h^\vee -1$ and $j=1,\ldots ,N$, implying that

$$\begin{aligned} X^i(j)\in \mathfrak {g}_1 \oplus \mathfrak {g}_2,\qquad i=1,\ldots ,h^\vee -1,\quad j=1,\ldots ,N. \end{aligned}$$

The space $\mathfrak {g}_1 \oplus \mathfrak {g}_2$ is a codimension 1 vector subspace of the symplectic space $\mathfrak {t}$ introduced in the previous section. As a consequence, the number of non-trivial unknowns reduces to $N\dim \mathfrak {t}=2 N (h^{\vee }-1)$. This fact represents a major advantage of working with the Gauge where all singularities are first order poles. Working for instance with opers in a canonical form (after fixing a transversal subspace $\mathfrak {s}$), then necessarily the singularities are higher order poles, and the total number of non-trivial unknowns we need to consider is again $(1+\frac{n h^{\vee }}{2} )N$—a number which grows quadratically with n—rather than $2 N(h^{\vee }-1)$, which grows linearly.

The trivial monodromy conditions (8.6) for the operator (8.5) are written in terms of the gradation (7.9). This gradation depends on the coefficients $\eta _1,\eta _2$, which are unknowns of our problem. In order to be able to derive an explicit system of equations for these unknowns, we write conditions (8.6) with respect to the fixed gradation (7.2), namely the gradation induced by the highest root. From here below we restrict to operators with local expansion (8.5) such that $a \in f+\mathfrak {b}_+$, $b \in \mathfrak {b}_+$, for this is the case of the Quantum $\widehat{\mathfrak {g}}$-KdV oper (6.1). From now on $\mathfrak {g}\ne \mathfrak {sl}_2$, and the $\mathfrak {sl}_2$ case in treated in a separate section of the paper.

Lemma 8.6

Let $\mathcal {L}\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\mathbb {P}^1})$ be given by (8.5), with $a \in f+\mathfrak {b}^+$, $b \in \mathfrak {b}^+$. Moreover, for $j\in \mathbb {Z}$ set $a_j=\pi _j(a)$, $b_j=\pi _j(b)$, $j\in \mathbb {Z}$, where $\pi _j$ is the projection defined in (7.3). If $\mathfrak {g}$ is not of type $\mathfrak {sl}_2$, then $a_{-2}=b_{-2}=b_{-1}=0$.

Proof

We have $b\in \mathfrak {b}\subset \mathfrak {g}_0\oplus \mathfrak {g}_1\oplus \mathfrak {g}_2$, which implies that $b_{-1}=b_{-2}=0$, while $a\in f+\mathfrak {b}\subset \mathfrak {g}_{-1}\oplus \mathfrak {g}_0\oplus \mathfrak {g}_1\oplus \mathfrak {g}_2$, implying $a_{-2}=0$. $\quad \square$

We can now write the trivial monodromy conditions (8.6) in terms of the gradation (7.2).

Proposition 8.7

Let $\mathcal {L}\in {{\,\mathrm{op}\,}}_{\mathfrak {g}}(K_{\mathbb {P}^1})$ be given by (8.5), with $a \in f+\mathfrak {b}_+$, $b \in \mathfrak {b}_+$. If $\mathfrak {g}$ is not of type $\mathfrak {sl}_2$, then the trivial monodromy conditions (8.6) are equivalent to the following system of equations for $\eta \in \mathfrak {n}_+$, $a\in f+\mathfrak {b}_+$, $b\in \mathfrak {b}_+$:

$$\begin{aligned}&\eta _0=0 \end{aligned}$$

(8.15a)

$$\begin{aligned}&2a_1-2[\eta _1, a_0]+[\eta _1,[\eta _1,a_{-1}]]-[\eta _2,a_{-1}]=0, \end{aligned}$$

(8.15b)

$$\begin{aligned}&2b_2+[\eta _1,[\eta _1,b_0]]-2[\eta _1,b_{1}]-[\eta _2,b_0]\nonumber \\&\,\,\, -\left[ 2a_2-[\eta _2,a_0]+\frac{1}{3}[\eta _1,[\eta _1,a_0]+2[\eta _2,a_{-1}]-4a_1],a_0-[\eta _1,a_{-1}]\right] =0, \end{aligned}$$

(8.15c)

where $\eta _i=\pi _i(\eta ),\,a_i=\pi _i(a),\,b_i=\pi _i(b)\in \mathfrak {g}_i$.

Proof

Due to Lemma 8.6 we have $a_{-2}=0$ and plugging $x=a$ in the relations (7.11) we get

$$\begin{aligned}&\pi ^R_{2}(a)=0\\&\pi ^R_{1}(a)=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}a_{-1}\nonumber \\&\pi ^R_{0}(a)=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}(a_0-{{\,\mathrm{ad}\,}}_{\eta _1}a_{-1})\nonumber \\&\pi ^R_{-1}(a)=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}(a_1+\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _1}^2 a_{-1}-\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _2} a_{-1}-{{\,\mathrm{ad}\,}}_{\eta _1} a_0)\\&\pi ^R_{-2}(a)=a_2-\frac{1}{6}{{\,\mathrm{ad}\,}}_{\eta _1}^3 a_{-1}+\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _1}^2 a_0-\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _2} a_0+\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _1}{{\,\mathrm{ad}\,}}_{\eta _2}a_{-1}-{{\,\mathrm{ad}\,}}_{\eta _1} a_1. \end{aligned}$$

On the other hand, after Lemma 8.6 we have $b_{-1}=b_{-2}=0$, so that for $x=b$ the relations (7.11) become $\pi ^R_{2}(b)=\pi ^R_{1}(b)=0$, and

$$\begin{aligned}&\pi ^R_{0}(b)=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}b_0\nonumber \\&\pi ^R_{-1}(b)=e^{{{\,\mathrm{ad}\,}}(\eta _1+\frac{1}{2}\eta _2)}(b_1-{{\,\mathrm{ad}\,}}_{\eta _1} b_0)\\&\pi ^R_{-2}(b)=b_2+\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _1}^2 b_0-\frac{1}{2}{{\,\mathrm{ad}\,}}_{\eta _2} b_0-{{\,\mathrm{ad}\,}}_{\eta _1} b_1. \end{aligned}$$

Plugging these formulae into (8.6), one gets (8.15). $\quad \square$

10 Trivial Monodromy for Quantum-KdV Opers

In this section we address the final assumption, namely Assumption 4, on quantum $\mathfrak {g}$-KdV opers: for any value of the loop algebra parameter $\lambda$, the monodromy at any regular non-zero singular point must be trivial. As a result, we completely characterise quantum $\mathfrak {g}$-KdV opers by means of a system of rational equations, see Proposition 9.2.

We thus consider an oper of the form (6.1) such that the set J of additional singularities is non-empty, namely $J=\lbrace 1,\ldots , N\rbrace$, for some $N\in \mathbb {Z}_+$. Hence (6.1) reads

$$\begin{aligned} \mathcal {L}=\partial _z+f+\frac{r}{z}+(z^{1-h^\vee }+\lambda z^k)E_{\theta }+\sum _{j=1}^{N}\frac{1}{z-w_j} \left( -\theta ^\vee +\sum _{i=1}^{h^\vee \!-1}\frac{X^i{(j)}}{z^i}\right) , \end{aligned}$$

(9.1)

where we put, and use from now on for convenience,

$$\begin{aligned} k=1-h^\vee -\hat{k}\in (-h^\vee ,1-h^\vee ) \end{aligned}$$

(9.2)

in place of $\hat{k}\in (0,1)$. In formula (9.1), the quantities $r\in \mathfrak {h}$ and $k\in (-h^\vee ,1-h^\vee )$ are given, $\lambda \in \mathbb {C}$ is arbitrary, while the non-zero pairwise distinct complex numbers $w_j$, and the Lie algebra elements $X^i(j)\in \mathfrak {g}^i$ are to be determined by the trivial monodromy conditions.

For any fixed $\ell =1,\ldots ,N$, the localization of $\mathcal {L}$ at $z=w_\ell$ yields an expansion of the form (8.5). Indeed, using $x=z-w_\ell$ as local coordinate, we get

$$\begin{aligned} \partial _x+ \frac{R(\ell )}{x}+a(\ell )+b(\ell )x+O(x^2), \end{aligned}$$

(9.3)

where the coefficients $R(\ell )=-\theta ^\vee +\eta (\ell )$ with $\eta (\ell )\in \mathfrak {n}_+$, $a(\ell )\in f+\mathfrak {b}_+$ and $b(\ell )\in \mathfrak {b}_+$ can be obtained from (9.1). Since $\mathcal {L}$ is of type (8.5), the trivial monodromy conditions at $z=w_\ell$ are provided by Proposition 8.7. Imposing that those trivial monodromy conditions are fulfilled for any value of $\lambda$, and using the expression of $\eta (\ell ), a(\ell ), b(\ell )$ in terms of the coefficients of (9.1), we obtain below a complete set of equations for the unknowns $w_j,X^i(j),j=1,\ldots ,N$.

10.1 Vector notation

In order to deal with all singularities $\{w_j,j=1,\ldots ,N\}$ at once, it will be useful to consider the following construction. For every pair of vectors $\mathbf{v}=(v_1,\ldots ,v_N)$ and $\mathbf{v}'=(v'_1,\ldots ,v'_N)$ in $\mathbb {C}^N$, denote by $\mathbf{v}\circ \mathbf{v}'$ the product algebra:

$$\begin{aligned} \mathbf{v}\circ \mathbf{v}'=(v_1v'_1,\ldots ,v_Nv'_N)\in \mathbb {C}^N, \end{aligned}$$

and extend the Lie algebra structure from $\mathfrak {g}$ to the tensor product $\mathbb {C}^N\!\otimes \mathfrak {g}$ by letting $[\mathbf{v}\otimes x,\mathbf{v}'\otimes y]=(\mathbf{v}\circ \mathbf{v}')\otimes [x,y]$, for $\mathbf{v},\mathbf{v}'\in \mathbb {C}$ and $x,y\in \mathfrak {g}$. Setting $\mathbf{1}=(1,\ldots ,1)\in \mathbb {C}^N$, then we have an injective homomorphism of Lie algebras $\mathfrak {g}\hookrightarrow \mathbb {C}^N\otimes \mathfrak {g}$ given by $x\mapsto \mathbf{1}\otimes x$, $x\in \mathfrak {g}$. By abuse of notation we denote in the same way elements of $\mathfrak {g}$ and their images in $\mathbb {C}^N\!\otimes \mathfrak {g}$ under this homomorphism. The elements $X^i(j)$ appearing in (9.1) can now be written in the more compact form

$$\begin{aligned} X^i=(X^i(1),\ldots ,X^i(N))\in \mathbb {C}^N\otimes \mathfrak {g}^i, \quad i=0,\ldots ,h^\vee -1. \end{aligned}$$

(9.4)

Moreover, for the additional poles $w_j$, $j=1,\ldots , N$ denote

$$\begin{aligned} \mathbf{w}=(w_1,\ldots ,w_n)\in \mathbb {C}^N, \end{aligned}$$

(9.5)

and for $s\in \mathbb {R}$ put $\mathbf{w}^s=(w^s_1,\ldots ,w^s_n)\in \mathbb {C}^N$. Let $\mathbb {C}(\mathbf{w})=\mathbb {C}(w_1,\ldots ,w_N)$ be the field of fractions of the polynomial ring $\mathbb {C}[\mathbf{w}]=\mathbb {C}[w_1,\ldots ,w_N]$. For $i\in \mathbb {Z}$, introduce the $N\times N$ matrices—with values in $\mathbb {C}(\mathbf{w})$—given by

$$\begin{aligned} (A_i)_{\ell j}= {\left\{ \begin{array}{ll} \frac{w_\ell }{w_\ell -w_j},&{} \quad \ell \ne j\\ -i&{} \quad \ell =j \end{array}\right. }, \qquad (B_i)_{\ell j}= {\left\{ \begin{array}{ll} (i+\frac{w_\ell }{w_\ell -w_j})\frac{w_\ell }{w_\ell -w_j},&{} \quad \ell \ne j\\ -\frac{i(i+1)}{2}&{} \quad \ell =j \end{array}\right. }, \end{aligned}$$

(9.6)

and define $A,B\in {{\,\mathrm{End}\,}}_{\mathbb {C}(\mathbf{w})}(\mathbb {C}^N\otimes \mathfrak {g})$ as

$$\begin{aligned} A(\mathbf{v}\otimes x)&=A_i(\mathbf{v})\otimes x,\qquad x\in \mathfrak {g}^i, \end{aligned}$$

(9.7)

$$\begin{aligned} B(\mathbf{v}\otimes x)&=B_i(\mathbf{v})\otimes x,\qquad x\in \mathfrak {g}^i \end{aligned}$$

(9.8)

In addition, introduce $M,S,Y\in {{\,\mathrm{End}\,}}_{\mathbb {C}(\mathbf{w})} (\mathbb {C}^N\otimes \mathfrak {g})$ as follows. For $X\in \mathbb {C}^N\otimes \mathfrak {g}$ let

$$\begin{aligned}&M(X)=A(X)-A_0(\mathbf{1})\circ X+[r,X], \end{aligned}$$

(9.9a)

$$\begin{aligned}&S(X)=2B(X)-B_0(\mathbf{1})\circ X+\frac{4}{3}k A(X)-\frac{k}{3}A_0(\mathbf{1})\circ X+(1+\frac{k}{3})[r,X], \end{aligned}$$

(9.9b)

$$\begin{aligned}&Y(X)=\frac{2}{3}k^2 X+\frac{2}{3}k A(X)+2B(X)+(1+\frac{k}{3})[r,X]. \end{aligned}$$

(9.9c)

We can now express the trivial monodromy conditions for the operator (9.1). Recall that

$$\begin{aligned} \pi _0(f)=-\sum _{i\in I_{\theta }}E_{-\alpha _i},\quad \pi _{-1}(f)=-\sum _{i\in I{\setminus } I_{\theta }}E_{-\alpha _i} \end{aligned}$$

(9.10)

are the projections (7.3) of the principal nilpotent element f with respect to the highest root gradation (7.2). The trivial monodromy conditions for the general case are expressed as follows:

Proposition 9.1

Let ${{\,\mathrm{rank}\,}}{\mathfrak {g}}>1$. The operator (9.1) has trivial monodromy at $z=w_\ell$ for every $\ell =1,\ldots ,N$ and every $\lambda \in \mathbb {C}$ if and only if the following conditions are satisfied:

• 1. For $i=0,\ldots ,h^\vee -1$, the variables $X^i$ belong to $\mathbb {C}^N\otimes \mathfrak {t}^i$, where $\mathfrak {t}=\mathbb {C}\theta ^\vee \oplus \mathfrak {g}_1 \oplus \mathfrak {g}_2\subset \mathfrak {b}^+$ is the symplectic vector space defined in (7.13), and $\mathfrak {t}^i=\mathfrak {t}\cap \mathfrak {g}^i$.

• 2. the set of variables $\{\mathbf{w}\}\cup \{X^i\,|\, i=1,\ldots ,h^\vee -1\}$ satisfies the following system of equations

$$\begin{aligned}&[[X^1,\pi _{-1}(f)],E_\theta ]=(k\mathbf{1}+\theta (r)\mathbf{1}-2A_0(\mathbf{1}))\otimes E_{\theta }, \end{aligned}$$

(9.11a)

$$\begin{aligned}&[X^{i+1},\pi _0(f)]=M(X^i)+\frac{1}{2}\sum _{s=0}^i[X^s,[X^{i+1-s},\pi _{-1}(f)]],\qquad i=1,\ldots ,h^\vee -2, \end{aligned}$$

(9.11b)

$$\begin{aligned}&2(1-h^\vee -k)\mathbf{w}\otimes E_\theta +Y(X^{h^\vee -1})+\sum _{i=1}^{h^\vee -1}[X^{h^\vee -1-i},S(X^i)]\nonumber \\&\quad =\frac{k}{3}\sum _{i=2}^{h^\vee -2}[X^i,[X^{h^\vee -i},\pi _0(f)]]. \end{aligned}$$

(9.11c)

Here, $X^0=-\mathbf{1}\otimes \theta ^\vee$, $E_\theta \in \mathfrak {g}$ is the highest root vector defined in Sect. 1.2, the operator $A_0\in {{\,\mathrm{End}\,}}_{\mathbb {C}(\mathbf{w})}(\mathbb {C}^N)$ is given in (9.6) and $M,S,Y\in {{\,\mathrm{End}\,}}_{\mathbb {C}(\mathbf{w})}(\mathbb {C}^N\otimes \mathfrak {g})$ are given in (9.9).

Proof

Recall that the monodromy about $z=w_\ell$ of an operator with expansion (9.3) is encoded in the quantities $R(\ell ),a(\ell ),b(\ell )$. These in turn can be expressed in terms of the variables $X^i,i=1\ldots h^{\vee }-1$. Defining $\mathbf{R}=(R(1),\ldots ,R(N)$, $\mathbf{a}=(a(1),\ldots a(N))$, $\mathbf{b}=(b(1),\ldots ,b(N))$ we have

$$\begin{aligned} \mathbf{R}&=-\mathbf{1}\otimes \theta ^\vee +{{\varvec{\eta }}},\qquad {{\varvec{\eta }}}=\sum _{i=1}^{h^\vee -1}{} \mathbf{w}^{-i}\circ X^i\in \mathbb {C}^N\otimes \mathfrak {g}.\nonumber \\ \mathbf{a}&=\mathbf{1}\otimes f+\mathbf{w}^{-1}\otimes r+(\mathbf{w}^{1-h^\vee }+ \lambda \mathbf{w}^k)\otimes E_{\theta }+\sum _{i=0}^{h^{\vee }-1}{} \mathbf{w}^{-i-1}\circ A(X^i)\nonumber \\ \mathbf{b}&=-\mathbf{w}^{-2}\otimes r+((1-h^\vee )\mathbf{w}^{-h^\vee }+k\lambda \mathbf{w}^{k-1}) \otimes E_{\theta }-\sum _{i=0}^{h^\vee -1}{} \mathbf{w}^{-i-2}\circ B(X^i). \end{aligned}$$

(9.12)

Note that the projections (7.3) onto the eigenspaces of ${{\,\mathrm{ad}\,}}_{\theta ^\vee }$ can be extended uniquely to $\mathbb {C}^N\otimes \mathfrak {g}$ by the rule $\pi _i(\mathbf{v}\otimes x)=\mathbf{v}\otimes \pi _i(x)$, with $\mathbf{v}\in \mathbb {C}^N$ and $x\in \mathfrak {g}$. From (8.15), the trivial monodromy conditions at all points $w_j$, $j=1,\ldots ,N$ can thus be written in the following compact form

$$\begin{aligned}&{{\varvec{\eta }}}_0=0, \end{aligned}$$

(9.13a)

$$\begin{aligned}&2\mathbf{a}_1-2[{{\varvec{\eta }}}_1,\mathbf{a}_0]+[{{\varvec{\eta }}}_1,[{{\varvec{\eta }}}_1,\mathbf{a}_{-1}]]-[{{\varvec{\eta }}}_2,\mathbf{a}_{-1}]=0, \end{aligned}$$

(9.13b)

$$\begin{aligned}&2\mathbf{b}_2+[{{\varvec{\eta }}}_1,[{{\varvec{\eta }}}_1,\mathbf{b}_0]]-2[{{\varvec{\eta }}}_1,\mathbf{b}_{1}]-[{{\varvec{\eta }}}_2,\mathbf{b}_0]\nonumber \\&\,\,\, =\left[ 2\mathbf{a}_2-[{{\varvec{\eta }}}_2,\mathbf{a}_0]+\frac{1}{3}[{{\varvec{\eta }}}_1,[{{\varvec{\eta }}}_1,\mathbf{a}_0]+2[{{\varvec{\eta }}}_2,\mathbf{a}_{-1}]-4\mathbf{a}_1],\mathbf{a}_0-[{{\varvec{\eta }}}_1,\mathbf{a}_{-1}]\right] . \end{aligned}$$

(9.13c)

Recall that by definition $X^i\in \mathbb {C}^N\otimes \mathfrak {g}^i$, and that $X^0=-\mathbf{1}\otimes \theta ^\vee \in \mathbb {C}^N\otimes \mathfrak {t}^0$. Due to (9.13a) then from (9.12) we obtain $\pi _0(X^i)=0$ for $i=1,\ldots , h^\vee -1$, and—by the definition of $\mathfrak {t}$—this implies $X^i\in \mathbb {C}^N\otimes \mathfrak {t}^i$, proving part (1). In particular, we obtain:

$$\begin{aligned} {{\varvec{\eta }}}_1=\sum _{i=1}^{h^\vee -2}{} \mathbf{w}^{-i}\circ X^i,\qquad {{\varvec{\eta }}}_2=\mathbf{w}^{1-h^\vee }\circ X^{h^\vee -1}. \end{aligned}$$

To prove part (2), we consider (9.13b) and (9.13c). First, note that we have $\mathbf{a}_{-2}=0$, while

$$\begin{aligned}&\mathbf{a}_{-1}=\mathbf{1}\otimes \pi _{-1}(f),\\&\mathbf{a}_0=\mathbf{1}\otimes \pi _{0}(f)+\mathbf{w}^{-1}\circ (\mathbf{1}\otimes r-A(\mathbf{1}\otimes \theta ^\vee )),\\&\mathbf{a}_1=\sum _{i=1}^{h^\vee -2}{} \mathbf{w}^{-i-1}\circ A(X^i),\\&\mathbf{a}_2=(\mathbf{w}^{1-h^\vee }+\lambda \mathbf{w}^k)\otimes E_{\theta }+\mathbf{w}^{-h^\vee }\circ A(X^{h^\vee -1}). \end{aligned}$$

On the other hand, $\mathbf{b}_{-2}=0$, $\mathbf{b}_{-1}=0$, and

$$\begin{aligned} \mathbf{b}_0&=\mathbf{w}^{-2}\circ (-\mathbf{1}\otimes r+B(\mathbf{1}\otimes \theta ^\vee )),\\ \mathbf{b}_1&=-\sum _{i=1}^{h^\vee -2}{} \mathbf{w}^{-i-2}\circ B(X^i),\\ \mathbf{b}_2&=((1-h^\vee )\mathbf{w}^{-h^\vee }+k\lambda \mathbf{w}^{k-1})\otimes E_{\theta }-\mathbf{w}^{-h^\vee -1}\circ B(X^{h^\vee -1}). \end{aligned}$$

Plugging the above quantities into (9.13b) we obtain

$$\begin{aligned} \sum _{i=1}^{h^\vee -2}{} \mathbf{w}^{-i-1}\circ \left( -[X^{i+1},\pi _{0}(f)]+M(X^i)+\frac{1}{2}\sum _{s=0}^i[X^s,[X^{i+1-s},\pi _{-1}(f)]]\right) =0. \end{aligned}$$

Since for each $i=1,\ldots ,h^\vee -2$ the element in the sum above which is multiplied by $\mathbf{w}^{-i-1}$ belongs to $\mathbb {C}^N\otimes \mathfrak {g}^i$, and since each component of $\mathbf{w}\in \mathbb {C}^N$ is different from zero, we get (9.11b). Now we consider (9.13c), which is linear with respect to $\lambda$. The vanishing of the coefficient of order 1 in $\lambda$ reads

$$\begin{aligned} k\mathbf{w}^{-1}\otimes E_\theta =[\mathbf{1 }\otimes E_\theta ,\mathbf{a}_0-[{{\varvec{\eta }}}_1,\mathbf{a}_{-1}]], \end{aligned}$$

(9.14)

which is equivalent to (9.11a). Note that from (9.14) it follows that for every $\mathbf{Q}\in \mathbb {C}^N\otimes \mathfrak {g}_2=\mathbb {C}^N\otimes \mathfrak {g}^{h^\vee -1}$ one has

$$\begin{aligned}{}[\mathbf{Q},\mathbf{a}_0-[{{\varvec{\eta }}}_1,\mathbf{a}_{-1}]]=k\mathbf{w }^{-1}\circ \mathbf{Q}. \end{aligned}$$

(9.15)

We now consider the vanishing of the coefficient of order zero in $\lambda$ in (9.13c). Since the term $2\mathbf{a}_2-[{{\varvec{\eta }}}_2,\mathbf{a}_0]+\frac{1}{3}[{{\varvec{\eta }}}_1,[{{\varvec{\eta }}}_1,\mathbf{a}_0]+2[{{\varvec{\eta }}}_2,\mathbf{a}_{-1}]-4\mathbf{a}_1]$ belongs to $\mathbb {C}^N\otimes \mathfrak {g}_2$ then using (9.15) we get that (9.13c) can be written as $2\mathbf{b}_2+[{{\varvec{\eta }}}_1,[{{\varvec{\eta }}}_1,\mathbf{b}_0]]-2[{{\varvec{\eta }}}_1,\mathbf{b}_{1}]-[{{\varvec{\eta }}}_2,\mathbf{b}_0]=k\mathbf{w }^{-1}\circ (2\mathbf{a}_2-[{{\varvec{\eta }}}_2,\mathbf{a}_0]+\frac{1}{3}[{{\varvec{\eta }}}_1,[{{\varvec{\eta }}}_1,\mathbf{a}_0]+2[{{\varvec{\eta }}}_2,\mathbf{a}_{-1}]-4\mathbf{a}_1])$. By a direct computation, the latter identity is shown to be equivalent to (9.11c). Part (2) of the proposition is proved. $\quad \square$

10.2 Trivial monodromy: system in $\mathbb {C}^{2N(h^{\vee }-1)}$

The trivial monodromy conditions (9.11) for the operator (9.1) are a system of equations in $\mathbb {C}^N\otimes \mathfrak {n}^+$. Due to Proposition 9.1(2), the variables $X^i,i\ge 1$ defined in (9.4) belong to $\mathbb {C}^N\otimes \mathfrak {t}^i$, where $\mathfrak {t}$ is the symplectic vector space defined in (7.13) and $\mathfrak {t}^i=\mathfrak {t}\cap \mathfrak {g}^i$. As it was already remarked, this implies the the total number of non-trivial variables $\lbrace \mathbf {w},X^i \rbrace$ is $2N(h^{\vee }-1)$. By choosing an explicit basis of $\mathfrak {t}$, we now write the system (9.11) as an equivalent system in $\mathbb {C}^{2N(h^{\vee }-1)}$.

Recall the set $\Theta \subset \mathfrak {h}^*$ defined in (7.15), and define

$$\begin{aligned} \Theta ^i=\{\alpha \in \Theta \,|\,{{\,\mathrm{ht}\,}}(\alpha )=i\}, \quad i=0,\ldots ,h^\vee -1. \end{aligned}$$

(9.16)

Recall the root vectors $\{E_\alpha , \alpha \in \Delta \}$ of $\mathfrak {g}$ defined in $\S$1.2. For $\alpha \in \Theta$, introduce the variables $\mathbf{x}^\alpha \in \mathbb {C}^N$, and write $X^i$ as

$$\begin{aligned} X^i=\sum _{\alpha \in \Theta ^i}{} \mathbf{x}^\alpha \otimes E_{\alpha },\qquad i=0,\ldots , h^\vee -1, \end{aligned}$$

(9.17)

with $\mathbf{x}^0=-\mathbf{1}$ and $E_0=\theta ^\vee$. Note that we always have $X^{h^\vee -1}=\mathbf{x}^\theta \otimes E_{\theta }$. For $\alpha \in \Theta$ define $M_\alpha ,S_\alpha , Y_\alpha \in {{\,\mathrm{End}\,}}_{\mathbb {C}(\mathbf{w})}(\mathbb {C}^N)$ as the unique linear operators satisfying the relations:

$$\begin{aligned} M_\alpha (\mathbf{v})\otimes E_\alpha&=M(\mathbf{v}\otimes E_\alpha ), \end{aligned}$$

(9.18a)

$$\begin{aligned} S_\alpha (\mathbf{v})\otimes E_\alpha&=S(\mathbf{v}\otimes E_\alpha ), \end{aligned}$$

(9.18b)

$$\begin{aligned} Y_\alpha (\mathbf{v})\otimes E_\alpha&=Y(\mathbf{v}\otimes E_\alpha ), \end{aligned}$$

(9.18c)

where $M,S,Y\in {{\,\mathrm{End}\,}}_{\mathbb {C}(\mathbf{w})}(\mathbb {C}^N\otimes \mathfrak {g})$ were introduced in (9.9). Explicitly, for $\mathbf{v}\in \mathbb {C}^N$ we have

$$\begin{aligned} M_\alpha (\mathbf{v})&=A_{{{\,\mathrm{ht}\,}}(\alpha )}(\mathbf{v})-A_0(\mathbf{1})\circ \mathbf{v}+\alpha (r)\mathbf{v}, \end{aligned}$$

(9.19)

$$\begin{aligned} S_\alpha (\mathbf{v})&=2B_{{{\,\mathrm{ht}\,}}(\alpha )}(\mathbf{v})+\frac{4}{3}kA_{{{\,\mathrm{ht}\,}}(\alpha )}(\mathbf{v})-B_0(\mathbf{1})\circ \mathbf{v}\nonumber \\&\quad -\frac{k}{3}A_0(\mathbf{1})\circ \mathbf{v} +(1+\frac{k}{3})\alpha (r)\mathbf{v}, \end{aligned}$$

(9.20)

$$\begin{aligned} Y_\alpha (\mathbf{v})&=\frac{2}{3}k^2\mathbf{v}+\frac{2}{3}k A_{{{\,\mathrm{ht}\,}}(\alpha )}(\mathbf{v})+ 2B_{{{\,\mathrm{ht}\,}}(\alpha )}(\mathbf{v})+(1+\frac{k}{3})\alpha (r)\mathbf{v}. \end{aligned}$$

(9.21)

with $\alpha \in \Theta$. Recall the bimultiplicative function $\epsilon _{\alpha ,\beta }$ ($\alpha ,\beta \in Q$) defined in (1.11), (1.12).

Proposition 9.2

Let ${{\,\mathrm{rank}\,}}\mathfrak {g}>1$, and let $A_0$ be given by (9.6) and $M_\alpha , S_\alpha , Y_\alpha$ by (9.18). Let the variables $\mathbf{x}^\alpha$, $\alpha \in \Theta$, be defined in terms of $X^i$, $i=0,\ldots ,h^\vee -1$ by (9.17).

The trivial monodromy conditions (9.11), which are a system of equations in the variables $\lbrace \mathbf {w},X^i \rbrace$, are equivalent to the following system of $2h^\vee -2$ ($\mathbb {C}^N$-valued) equations in the $2h^\vee -2$ ($\mathbb {C}^N$-valued) variables $\{\mathbf{w},\mathbf{x}^\alpha \,|\,\alpha \in \Theta \}$:

$$\begin{aligned} \sum _{\alpha \in \Theta ^1}{} \mathbf{x}^\alpha =(k+\theta (r))\mathbf{1}-2A_0(\mathbf{1}), \end{aligned}$$

(9.22a)

for every $\alpha \in \Theta ^i$, $i=1,\ldots ,h^\vee -3$:

$$\begin{aligned} \sum _{\begin{array}{c} j\in I_\theta :\\ \alpha +\alpha _j\in \Theta ^{i+1} \end{array}}\epsilon _{\alpha ,\alpha _j}\, \mathbf{x}^{\alpha +\alpha _j}&=M_\alpha (\mathbf{x}^\alpha )-\frac{1}{2}\sum _{\beta \in \Theta ^1}(\alpha \vert \beta )\, \mathbf{x}^\alpha \circ \mathbf{x}^{\beta }\nonumber \\ +&\frac{1}{2}\sum _{s=1}^{i-1}\sum _{\beta \in \Theta ^1}\sum _{\begin{array}{c} \gamma \in \Theta ^s: \\ \alpha -\gamma \in \Delta \\ \alpha -\gamma +\beta \in \Delta \end{array}}\epsilon _{\alpha ,\gamma } \epsilon _{\alpha ,\beta }\epsilon _{\gamma ,\beta }{} \mathbf{x}^\gamma \circ \mathbf{x}^{\alpha -\gamma +\beta }, \end{aligned}$$

(9.22b)

for every $\alpha \in \Theta ^{h^\vee -2}$:

$$\begin{aligned} \varepsilon _{\theta ,\alpha } \mathbf{x}^\theta&=2M_\alpha (\mathbf{x}^\alpha )-\sum _{\beta \in \Theta ^1}(\alpha \vert \beta ) \mathbf{x}^\alpha \circ \mathbf{x}^{\beta }\nonumber \\&+\sum _{s=1}^{h^\vee -3}\sum _{\beta \in \Theta ^1}\sum _{\begin{array}{c} \gamma \in \Theta ^s\\ \alpha -\gamma \in \Delta \\ \alpha -\gamma +\beta \in \Delta \end{array}}\epsilon _{\alpha ,\gamma }\epsilon _{\alpha ,\beta } \epsilon _{\gamma ,\beta }{} \mathbf{x}^\gamma \circ \mathbf{x}^{\alpha -\gamma +\beta }, \end{aligned}$$

(9.22c)

and finally

$$\begin{aligned} 2(k+h^\vee -1)\mathbf{w}&=Y_{\theta }(\mathbf{x}^\theta )-\sum _{i=1}^{h^\vee -1}\sum _{\alpha \in \Theta ^i}(\theta | \alpha ) \epsilon _{\theta ,\alpha }{} \mathbf{x}^{\theta -\alpha }\circ S_{\alpha }(\mathbf{x}^\alpha )\nonumber \\&-\frac{k}{3}\sum _{i=2}^{h^\vee -2}\sum _{j\in I_\theta }\sum _{\begin{array}{c} \gamma \in \Theta ^{i}\\ \theta -\gamma +\alpha _j\in \Delta \end{array}}\varepsilon _{\theta ,\gamma }\varepsilon _{\theta ,\alpha _j} \varepsilon _{\gamma ,\alpha _j}{} \mathbf{x}^{\theta -\gamma +\alpha _j}\circ \mathbf{x}^{\gamma }, \end{aligned}$$

(9.22d)

where $\mathbf{x}^0=-\mathbf{1}$.

Proof

System (9.22) is obtained substituting (9.17) into (9.11) and using the commutation relations (1.13). $\quad \square$

Remark 9.3

In the case $N=1$, namely in the case of a single additional singular pole $w_1$, system (9.22) greatly simplifies. Indeed, in this case the variables $\mathbf {x}^{\alpha }\in \mathbb {C}$ are scalars, and the operators $M_\alpha , S_\alpha ,Y_\alpha :\mathbb {C}\rightarrow \mathbb {C}$ are just multiplication operators—independent of the position of the additional pole $w_1$—and given by:

$$\begin{aligned} M_\alpha (\mathbf{v})&=\left( \alpha (r)-{{\,\mathrm{ht}\,}}(\alpha )\right) \mathbf{v},\nonumber \\ S_\alpha (\mathbf{v})&=\left( (1+\frac{k}{3})\alpha (r)-({{\,\mathrm{ht}\,}}(\alpha )+1+\frac{4}{3}k) {{\,\mathrm{ht}\,}}(\alpha )\right) \mathbf{v}\nonumber \\ Y_\alpha (\mathbf{v})&=\left( \frac{2}{3}k^2+(1+\frac{k}{3})\alpha (r)-({{\,\mathrm{ht}\,}}(\alpha )+1+\frac{2}{3}k) {{\,\mathrm{ht}\,}}(\alpha )\right) \mathbf{v}, \end{aligned}$$

(9.23)

for every $\mathbf{v}\in \mathbb {C}$. It follows that for $N=1$ the system (9.22) decouples: the first three equations are a subsystem for the $\mathbf{x}^{\alpha }$’s alone, and last equation yields the location of the pole $w_1$ as an explicit function of the $\mathbf{x}^{\alpha }$’s.

Let $P_n(N)$ be the number of n-coloured partitions of N. The Fock space of the quantum $\widehat{\mathfrak {g}}$-KdV model is generated by the action of $n={{\,\mathrm{rank}\,}}\mathfrak {g}$ free fields [29, 33]. Hence the number of the states of a given level N of the quantum theory is less or equal⁸ than $P_n(N)$ for arbitrary values of the parameters $(r,\hat{k})\in \mathfrak {h}\times (0,1)$ of the model, and it actually coincides with $P_n(N)$ for generic values of the parameters. According to Conjecture 0.1, the solutions of (9.22) are in bijections with the states of level N of quantum $\mathfrak {g}$-KdV. Hence, the ODE/IM conjecture implies the following conjecture on the number of solutions of (9.22).

Conjecture 9.4

The number of solutions of (9.22) is less or equal than $N!P_n(N)$. The set of parameters $(r,\hat{k})$ for which the number of solutions is $N!P_n(N)$ is a generic subset of $\mathfrak {h}\times (0,1)$.

The appearance of the factorial term N! in the conjecture above is due to the fact that solutions of system (9.22) differing only by a permutation of the additional poles $(w_1,\ldots ,w_N)$ give the same oper.

11 Explicit Computations

System (9.22), providing trivial monodromy conditions for the operator (9.1), is a system of $2N (h^{\vee }-1)$ equations in $2N (h^{\vee }-1)$ unknowns, which depends on the root structure of the algebra. In this section we provide an explicit presentation of this system in case $\mathfrak {g}=A_n, n \ge 2,D_n, n\ge 4$, and $E_6$. We omit to show our computations in the case $E_7,E_8$ due to their excessive length. In each case, we are able to reduce (9.22) to a system of 2N equations in 2N unknowns. Moreover, if $N=1$, we further reduce it to a single degree n polynomial equation in one variable. This is consistent with the ODE/IM hypothesis—see Conjecture 9.4—since the dimensions of the level 1 subspace of the quantum $\mathfrak {g}$-KdV model is equal to ${{\,\mathrm{rank}\,}}{g}=n$, for generic values of the central charge and of the vacuum parameters.

As already remarked in the Introduction, the $A_2$ case is now described in detail in [41].

11.1 The case $A_n$, $n\ge 2$

For $\mathfrak {g}$ of type $A_n$ we have $h^\vee =n+1$, $\theta =\sum _{i\in I}\alpha _i$, and $\dim \mathfrak {t}=2n$. Since $I{\setminus } I_\theta =\{1,n\}$ then from (9.10) we get

$$\begin{aligned} \pi _0(f)=-\sum _{i=2}^{n-1}E_{-\alpha _i},\quad \pi _{-1}(f)=-E_{-\alpha _1}-E_{-\alpha _n}. \end{aligned}$$

Defining

$$\begin{aligned} \beta _j=\sum _{i=n+1-j}^n\alpha _i,\quad \gamma _j=\sum _{i=1}^j\alpha _i, \qquad j=1,\ldots ,n-1 \end{aligned}$$

(10.1)

then we get $\Theta =\{0,\theta \}\cup \{\beta _i,\gamma _i,i=1,\ldots ,n-1\}$ and $\{E_{\alpha },\alpha \in \Theta \}$ is a basis of $\mathfrak {t}$, with $E_0=\theta ^\vee$. Thus, (9.17) reads

$$\begin{aligned} X^i= {\left\{ \begin{array}{ll} -\mathbf{1}\otimes \theta ^\vee &{} \quad i=0,\\ \mathbf{x}^{\beta _i}\otimes E_{\beta _i}+\mathbf{x}^{\gamma _i}\otimes E_{\gamma _i} &{} \quad i=1,\ldots ,n-1,\\ \mathbf{x}^\theta \otimes E_\theta &{} \quad i=n, \end{array}\right. } \end{aligned}$$

(10.2)

and system (9.22) takes the simpler form

$$\begin{aligned}&\mathbf{x}^{\beta _1}+\mathbf{x}^{\gamma _1}=(k+\theta (r))\mathbf{1}-2A_0(\mathbf{1}),\\&\mathbf{x}^{\beta _{i+1}}=M_{\beta _i}(\mathbf{x}^{\beta _i})-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _{i}}, i=1,\ldots ,n-2,\\&\mathbf{x}^{\gamma _{i+1}}=-M_{\gamma _i}(\mathbf{x}^{\gamma _i})+\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\gamma _{i}}, i=1,\ldots ,n-2,\\&\mathbf{x}^\theta =2M_{\beta _{n-1}}(\mathbf{x}^{\beta _{n-1}})+\sum _{s=1}^{n-1}{} \mathbf{x}^{\beta _s}\circ \mathbf{x}^{\gamma _{n-s}}-2\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _{n-1}}\\&-\mathbf{x}^\theta =2M_{\gamma _{n-1}}(\mathbf{x}^{\beta _{n-1}})+\sum _{s=1}^{n-1}{} \mathbf{x}^{\gamma _s}\circ \mathbf{x}^{\beta _{n-s}}-2\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\gamma _{n-1}}\\&2(n+k)\mathbf{w}=Y_\theta (\mathbf{x}^\theta )+\sum _{i=1}^{n-1}\left( \mathbf{x}^{\beta _{n-i}}\circ S_{\gamma _i}(\mathbf{x}^{\gamma _i})-\mathbf{x}^{\gamma _{n-i}}\circ S_{\beta _i}(\mathbf{x}^{\beta _i})\right) \\&\quad -2S_\theta (\mathbf{x}^\theta )+\frac{2}{3}k\sum _{i=1}^{n-2}{} \mathbf{x}^{\beta _{n-i}}\circ \mathbf{x}^{\gamma _{i+1}}. \end{aligned}$$

The above system can be further simplified as follows. Introduce the polynomial functions $P_i,\widetilde{P}_i:\mathbb {C}^N\rightarrow \mathbb {C}^N$, depending on the parameters $\mathbf{w}\in \mathbb {C}^N$ and defined recursively by the relations

$$\begin{aligned} P_{i+1}(\mathbf{x})&=M_{\beta _i}(P_i(\mathbf{x}))-\mathbf{x}\circ P_i(\mathbf{x}), \end{aligned}$$

(10.3a)

$$\begin{aligned} \widetilde{P}_{i+1}(\mathbf{x})&=-M_{\gamma _i}(\widetilde{P}_i(\mathbf{x}))+\mathbf{x}\circ \widetilde{P}_i(\mathbf{x}), \end{aligned}$$

(10.3b)

for every $\mathbf{x}\in \mathbb {C}^N$, and with $P_0(\mathbf{x})=-\mathbf{1},$$\widetilde{P}_0(\mathbf{x})=\mathbf{1},$ and $\beta _0=\gamma _0=0$.

Proposition 10.1

Let $\mathfrak {g}$ be of type $A_n$, $n\ge 2$. The operator (9.1) with $X^i$ as in (10.2) has trivial monodromy at all $w_{\ell }$, $\ell =1,\ldots ,N$ for all values of $\lambda$ if and only if:

• 1. The variables $\mathbf{x}^{\beta _1},\mathbf{x}^{\gamma _1},\mathbf{w} \in \mathbb {C}^N$ satisfy the system

  $$\begin{aligned}&\mathbf{x}^{\beta _1}+\mathbf{x}^{\gamma _1}=(k+\theta (r))\mathbf{1}-2A_0(\mathbf{1}),&\end{aligned}$$

  (10.4a)
  $$\begin{aligned}&\sum _{s=0}^n P_s(\mathbf{x}^{\beta _1})\circ \widetilde{P}_{n-s}(\mathbf{x}^{\gamma _1})=0 \end{aligned}$$

  (10.4b)
  $$\begin{aligned} 2(n+k)\mathbf{w}&=Y_\theta (P_n(\mathbf{x}^{\beta _1})+\widetilde{P}_n(\mathbf{x}^{\gamma _1}))+ \frac{2}{3}k\sum _{i=1}^{n-2}P_{n-i}(\mathbf{x}^{\beta _{1}})\circ \widetilde{P}_{i+1}(\mathbf{x}^{\gamma _{1}})\nonumber \\&+\sum _{i=1}^{n}\left( P_{n-i}(\mathbf{x}^{\beta _{1}})\circ S_{\gamma _i}(\widetilde{P}_i(\mathbf{x}^{\gamma _1})) -\widetilde{P}_{n-i}(\mathbf{x}^{\gamma _{1}})\circ S_{\beta _i}(P_i(\mathbf{x}^{\beta _1}))\right) , \end{aligned}$$

  (10.4c)

  where $\gamma _n=\beta _n=\theta$.

• 2. The variables $\mathbf{x}^\alpha \in \mathbb {C}^N$, $\alpha \in \Theta$, are given in terms of $\mathbf{x}^{\beta _1},\mathbf{x}^{\gamma _1}$ as

  $$\begin{aligned}&\mathbf{x}^{\beta _i}=P_i(\mathbf{x}^{\beta _1})&i=1,\ldots ,n-1,\\&\mathbf{x}^{\gamma _i}=\widetilde{P}_i(\mathbf{x}^{\gamma _1})&i=1,\ldots ,n-1,\\&\mathbf{x}^\theta =P_n(\mathbf{x}^{\beta _1})+\widetilde{P}_n(\mathbf{x}^{\gamma _1}), \end{aligned}$$

Corollary 10.2

Let $N=1$. The system (10.4) admits, for generic values of $r \in \mathfrak {h},k \in \mathbb {R}$, n solutions.

Proof

For $N=1$, the recursion relations (10.3) can be explicitely solved. Indeed, in this case we have $\mathbf{x}^\alpha =x^\alpha \in \mathbb {C}$ and the operator (9.19) reduces to the scalar operator $M_\alpha (x)=(\alpha (r)-{{\,\mathrm{ht}\,}}(\alpha ))x$, with $x\in \mathbb {C}$ and $\alpha \in \Theta$. Noting that ${{\,\mathrm{ht}\,}}(\beta _i)={{\,\mathrm{ht}\,}}(\gamma _i)=i$, then the polynomials

$$\begin{aligned} P_i(x)&=(-1)^{i+1}\prod _{j=1}^{i}(x-\beta _{j-1}(r)+j-1), \end{aligned}$$

(10.5a)

$$\begin{aligned} \widetilde{P}_i(x)&=\prod _{j=1}^{i}(x-\gamma _{j-1}(r)+j-1) \end{aligned}$$

(10.5b)

satisfy (10.3) with $P_0(x)=-1$ and $\widetilde{P}_0(x)=1$ (and $\beta _0=\gamma _0=0$). Since the polynomials $P,\widetilde{P}$ do not depend on the pole $w_1$, the system (10.4) splits into a subsystem for $x^{\beta _1},x^{\gamma _1}$ and a linear equation for $\mathbf{w}=w_1$, the additional pole,. Explicitly we have:

$$\begin{aligned}&x^{\beta _1}+x^{\gamma _1}=k+\theta (r),\\&\quad \sum _{s=0}^n (-1)^{s+1}\prod _{j=1}^{s}(x^{\beta _1}-\beta _{j-1}(r)+j-1) \prod _{j=1}^{n-s}(x^{\gamma _1}-\gamma _{j-1}(r)+j-1) =0. \end{aligned}$$

Substituting the first equation in the second, one obtain a polynomial equation for the variable $x^{\beta _1}$, which has—for generic values of r and k—n distinct solutions. Once a solution of the above system is chosen, the additional pole is given by

$$\begin{aligned} 2&(n+k)w_1=\frac{2}{3}k(k+n)\left( (-1)^{n+1}\prod _{j=1}^{n}(x^{\beta _1} -\beta _{j-1}(r)+j-1)+\prod _{j=1}^{n}(x^{\gamma _1}-\gamma _{j-1}(r)+j-1)\right) \\&+\frac{2}{3}k\sum _{i=1}^{n-2}\left( (-1)^{n-i+1}\prod _{j=1}^{n-i}(x^{\beta _1} -\beta _{j-1}(r)+j-1)\prod _{j=1}^{i+1}(x^{\gamma _1}-\gamma _{j-1}(r)+j-1)\right) \\&+\sum _{i=1}^{n-1}m_{n,i,r,k}\left( (-1)^{n-i+1}\prod _{j=1}^{n-i}(x^{\beta _1} -\beta _{j-1}(r)+j-1)\prod _{j=1}^{i}(x^{\gamma _1}-\gamma _{j-1}(r)+j-1)\right) , \end{aligned}$$

with $m_{n,i,r,k}=(1+\frac{k}{3})(\gamma _i(r)-\beta _{n-i}(r))+(n+1+\frac{4}{3}k)(n-2i)$, and

$$\begin{aligned}&x^{\beta _i}=(-1)^{i+1}\prod _{j=1}^{i}(x^{\beta _1}-\beta _{j-1}(r)+j-1),\qquad i=1,\ldots ,n-1,\\&x^{\gamma _i}=\prod _{j=1}^{i}(x^{\gamma _1}-\gamma _{j-1}(r)+j-1),\qquad i=1,\ldots ,n-1,\\&x^\theta =(-1)^{n+1}\prod _{j=1}^{n}(x^{\beta _1}-\beta _{j-1}(r)+j-1)+\prod _{j=1}^{n}(x^{\gamma _1}-\gamma _{j-1}(r)+j-1). \end{aligned}$$

$\square$

11.2 The case $D_n$

For $\mathfrak {g}$ of type $D_n$ we have $h^\vee =2n-2$, $\theta =\alpha _1+2\sum _{i=2}^{n-2}\alpha _i+\alpha _{n-1}+\alpha _n$, and $\dim \mathfrak {t}=4n-6$. Since $I{\setminus } I_\theta =\{2\}$ then from (9.10) we get

$$\begin{aligned} \pi _0(f)=-E_{-\alpha _1}-\sum _{i=3}^{n}E_{-\alpha _i},\quad \pi _{-1}(f)=-E_{-\alpha _2}. \end{aligned}$$

Denoting the roots

$$\begin{aligned}&\beta _j= {\left\{ \begin{array}{ll} \sum _{i=1}^{j+1}\alpha _i,&{} \quad j=1,\ldots ,n-3,\\ \sum _{i=1}^{2n-j-3}\alpha _i+2\sum _{i=2n-j-2}^{n-2}\alpha _i+\alpha _{n-1}+\alpha _n,&{} \quad j=n-1,\ldots ,2n-5,\\ \end{array}\right. } \\&\beta _{n-2}^+=\sum _{i=2}^{n-1}\alpha _i,\qquad \beta _{n-2}^-=\sum _{i=2}^{n-2}\alpha _i+\alpha _n, \end{aligned}$$

and

$$\begin{aligned}&\gamma _j= {\left\{ \begin{array}{ll} \sum _{i=2}^{j+1}\alpha _i,&{} \quad j=1,\ldots ,n-3,\\ \sum _{i=2}^{2n-j-3}\alpha _i+2\sum _{i=2n-j-2}^{n-2}\alpha _i+\alpha _{n-1}+\alpha _n,&{} \quad j=n-1,\ldots ,2n-5,\\ \end{array}\right. } \\&\gamma _{n-2}^+=\sum _{i=1}^{n-2}\alpha _i+\alpha _n,\qquad \gamma _{n-2}^-=\sum _{i=1}^{n-1}\alpha _i, \end{aligned}$$

then we have $\Theta =\{0,\beta ^\pm _{n-2},\gamma ^\pm _{n-2},\theta \}\cup \{\beta _j,\gamma _j,j=1,\ldots ,\widehat{n-2},\ldots ,2n-5\}$. Thus, (9.17) reads

$$\begin{aligned} X^i= {\left\{ \begin{array}{ll} -\mathbf{1}\otimes \theta ^\vee &{}\quad i=0,\\ \mathbf{x}^{\gamma _1}\otimes E_{\gamma _1} &{}\quad i=1,\\ \mathbf{x}^{\beta _{i-1}}\otimes E_{\beta _{i-1}}+\mathbf{x}^{\gamma _i}\otimes E_{\gamma _i} &{}\quad i=2,\ldots ,n-3,\\ \mathbf{x}^{\beta _{n-3}}\otimes E_{\beta _{n-3}}+\mathbf{x}^{\beta ^+_{n-2}}\otimes E_{\beta ^+_{n-2}}+\mathbf{x}^{\beta ^-_{n-2}}\otimes E_{\beta ^-_{n-2}}&{}\quad i=n-2\\ \mathbf{x}^{\gamma _{n-1}}\otimes E_{\gamma _{n-1}}+\mathbf{x}^{\gamma ^+_{n-2}}\otimes E_{\gamma ^+_{n-2}}+\mathbf{x}^{\gamma ^-_{n-2}}\otimes E_{\gamma ^-_{n-2}}&{}\quad i=n-1\\ \mathbf{x}^{\beta _{i-1}}\otimes E_{\beta _{i-1}}+\mathbf{x}^{\gamma _i}\otimes E_{\gamma _i} &{}\quad i=n,\ldots ,2n-5,\\ \mathbf{x}^{\beta _{2n-5}}\otimes E_{\beta _{2n-5}} &{}\quad i=2n-4,\\ \mathbf{x}^\theta \otimes E_\theta &{}\quad i=2n-3. \end{array}\right. } \end{aligned}$$

(10.6)

and system (9.22) takes the form

$$\begin{aligned}&\mathbf{x}^{\gamma _1}=(k+\theta (r))\mathbf{1}-2A_0(\mathbf{1}),&\\&\mathbf{x}^{\beta _{i}}-\mathbf{x}^{\gamma _{i+1}}=M_{\gamma _i}(\mathbf{x}^{\gamma _i})-\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\gamma _{i}},& i=1,\ldots ,n-4,\\&-\mathbf{x}^{\beta _{i}}=M_{\beta _{i-1}}(\mathbf{x}^{\beta _{i-1}})-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\gamma _{i-1}},& i=2,\ldots ,n-3,\\&\mathbf{x}^{\beta _{n-3}}-\mathbf{x}^{\beta ^+_{n-2}}-\mathbf{x}^{\beta ^-_{n-2}}=M_{\gamma _{n-3}}(\mathbf{x}^{\gamma _{n-3}})-\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\gamma _{n-3}}\\&\mathbf{x}^{\gamma ^+_{n-2}}-\mathbf{x}^{\gamma _{n-1}}=M_{\beta ^-_{n-2}}(\mathbf{x}^{\beta ^-_{n-2}})-\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\beta ^-_{n-2}}\\&\mathbf{x}^{\gamma ^-_{n-2}}-\mathbf{x}^{\gamma _{n-1}}=M_{\beta ^+_{n-2}}(\mathbf{x}^{\beta ^+_{n-2}})-\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\beta ^+_{n-2}}\\&-\mathbf{x}^{\gamma ^+_{n-2}}-\mathbf{x}^{\gamma ^-_{n-2}}=M_{\beta _{n-3}}(\mathbf{x}^{\beta _{n-3}})-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\gamma _{n-3}}\\&-\mathbf{x}^{\beta _{n-1}}=M_{\gamma ^+_{n-2}}(\mathbf{x}^{\gamma ^+_{n-2}})-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta ^-_{n-2}}\\&-\mathbf{x}^{\beta _{n-1}}=M_{\gamma ^-_{n-2}}(\mathbf{x}^{\gamma ^-_{n-2}})-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta ^+_{n-2}}\\&\mathbf{x}^{\beta _{i}}+\mathbf{x}^{\gamma _{i+1}}=M_{\gamma _i}(\mathbf{x}^{\gamma _i})-\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\gamma _{i}},& i=n-1,\ldots ,2n-6,\\&\mathbf{x}^{\beta _{i}}=M_{\beta _{i-1}}(\mathbf{x}^{\beta _{i-1}})-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\gamma _{i-1}},& i=n,\ldots ,2n-5,\\&\mathbf{x}^{\beta _{2n-5}}=M_{\gamma _{2n-5}}(\mathbf{x}^{\gamma _{2n-5}})-\mathbf{x}^{\beta ^+_{n-2}}\circ \mathbf{x}^{\beta ^-_{n-2}}+\sum _{s=1}^{n-4}{} \mathbf{x}^{\gamma _{s+1}}\circ \mathbf{x}^{\gamma _{2n-5-s}},&\\&\mathbf{x}^\theta =2M_{\beta _{2n-5}}(\mathbf{x}^{\beta _{2n-5}})+\sum _{s=2}^{n-3}{} \mathbf{x}^{\beta _s}\circ \mathbf{x}^{\gamma _{2n-4-s}}+\sum _{s=n-1}^{2n-5}{} \mathbf{x}^{\beta _s}\circ \mathbf{x}^{\gamma _{2n-4-s}}\\&\quad -\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\gamma _{2n-5}}-\mathbf{x}^{\beta ^+_{n-2}}\circ \mathbf{x}^{\gamma ^+_{n-2}}-\mathbf{x}^{\beta ^-_{n-2}}\circ \mathbf{x}^{\gamma ^-_{n-2}},&\\&2(k+2n-3)\mathbf{w}=Y_\theta (\mathbf{x}^\theta )+\mathbf{x}^{\beta _{n-2}^+}\circ S_{\gamma _{n-2}^+}(\mathbf{x}^{\gamma _{n-2}^+})-\mathbf{x}^{\gamma _{n-2}^+}\circ S_{\beta _{n-2}^+}(\mathbf{x}^{\beta _{n-2}^+})&\\&\quad +\mathbf{x}^{\beta _{n-2}^-}\circ S_{\gamma _{n-2}^-}(\mathbf{x}^{\gamma _{n-2}^-})-\mathbf{x}^{\gamma _{n-2}^-}\circ S_{\beta _{n-2}^-}(\mathbf{x}^{\beta _{n-2}^-})+\frac{2}{3}k\mathbf{x}^{\gamma _{n-2}^+}\circ \mathbf{x}^{\gamma _{n-2}^-}&\\&\quad +\sum _{\begin{array}{c} i=1,\ldots ,2n-5\\ i\ne n-2 \end{array}}\left( \mathbf{x}^{\beta _{2n-4-i}}\circ \left( S_{\gamma _i}(\mathbf{x}^{\gamma _i})-\frac{k}{3} \mathbf{x}^{\beta _i}\right) -\mathbf{x}^{\gamma _{2n-4-i}}\circ S_{\beta _i}(\mathbf{x}^{\beta _i})\right)&\\&\quad +\frac{2}{3}k(\mathbf{x}^{\beta _{n-2}^+}+\mathbf{x}^{\beta _{n-2}^-})\circ \mathbf{x}^{\beta _{n-1}}-\frac{2}{3}k(\mathbf{x}^{\gamma _{n-2}^+}+\mathbf{x}^{\gamma _{n-2}^-})\circ \mathbf{x}^{\gamma _{n-1}}&\\&\quad -\frac{2}{3}k\sum _{i=2}^{n-3}(\mathbf{x}^{\gamma _{2n-3-i}}\circ \mathbf{x}^{\beta _{i}}-\mathbf{x}^{\beta _{2n-3-i}}\circ \mathbf{x}^{\gamma _{i}}).&\end{aligned}$$

The above system can be simplified as follows. For $\alpha \in \Theta$ and $\mathbf{v}\in \mathbb {C}^N$, denote by $\widetilde{M}_\alpha$ the operators

$$\begin{aligned} \widetilde{M}_\alpha (\mathbf{v})=M_\alpha (\mathbf{v})-(k+\theta (r))\mathbf{v}+2A_0(\mathbf{1})\circ \mathbf{v}, \end{aligned}$$

(10.7)

and introduce the polynomials $P_i,\widetilde{P}_i:\mathbb {C}^N\rightarrow \mathbb {C}^N$ defined by the recursion relations

$$\begin{aligned} {\left\{ \begin{array}{ll} \widetilde{P}_{i+1}(\mathbf{x})=P_i(\mathbf{x})-\widetilde{M}_{\gamma _i}(\widetilde{P}_i(\mathbf{x}))\\ P_{i+1}(\mathbf{x})=\mathbf{x}\circ \widetilde{P}_i(\mathbf{x})-M_{\beta _i}(P_i(\mathbf{x})), \end{array}\right. } \qquad i\ge 0, \end{aligned}$$

(10.8)

with $P_0(\mathbf{x})=0$, $\widetilde{P}_0(\mathbf{x})=\mathbf{1}$ and $\gamma _0=\beta _0=0$. (In particular, from (10.8) we obtain $\widetilde{P}_1(\mathbf{x})=(k+\theta (r))\mathbf{1}-2A_0(\mathbf {1})$ and $P_1(\mathbf{x})=\mathbf{x}$.) Set

$$\begin{aligned} \widetilde{M}_\pm =M_{\beta ^\pm _{n-2}},\qquad M_\pm =M_{\gamma ^\pm _{n-2}}, \end{aligned}$$

and consider the rational functions $R_\pm :\mathbb {C}^N\rightarrow \mathbb {C}^N$, depending parametrically on $\mathbf{w}\in \mathbb {C}^N$ and given by:

$$\begin{aligned} R_\pm (\mathbf{x})&=[(M_++M_-)(\widetilde{M}_++\widetilde{M}_-)-4\,\mathbf{x}\,\circ ]^{-1}[(M_++M_-)\widetilde{M}_\mp (\widetilde{P}_{n-2}(\mathbf{x}))\\&\quad -2\mathbf{x}\circ \widetilde{P}_{n-2}(\mathbf{x})\pm (M_++M_-)P_{n-2}(\mathbf{x})].\\ \widetilde{R}_{\pm }(\mathbf{x})&=\frac{1}{2}P_{n-2}(\mathbf{x})+\frac{1}{2}\widetilde{M}_\mp (R_{\mp }(\mathbf{x}))-\frac{1}{2}M_\pm (R_{\pm }(\mathbf{x})), \end{aligned}$$

with $\mathbf{x}\in \mathbb {C}^N$. Finally, introduce recursively the polynomials $J,\widetilde{J}:\mathbb {C}^N\rightarrow \mathbb {C}^N$ as

$$\begin{aligned} {\left\{ \begin{array}{ll} \widetilde{J}_{i+1}(\mathbf{x})=-J_i(\mathbf{x})+\widetilde{M}_{\gamma _i}(\widetilde{J}_i(\mathbf{x}))\\ J_{i+1}(\mathbf{x})=M_{\beta _i}(J_i(\mathbf{x}))-\mathbf{x}\circ \widetilde{J}_i(\mathbf{x}), \end{array}\right. } \qquad i\ge n-1, \end{aligned}$$

(10.9)

with

$$\begin{aligned} \widetilde{J}_{n-1}(\mathbf{x})&=\frac{1}{2}P_{n-2}(\mathbf{x})-\frac{1}{2}\widetilde{M}_+(R_+(\mathbf{x}))-\frac{1}{2}\widetilde{M}_-(R_-(\mathbf{x}))\\ J_{n-1}(\mathbf{x})&=\frac{1}{2}{} \mathbf{x}\circ (R_+(\mathbf{x})+R_-(\mathbf{x}))-\frac{1}{2}M_+(\widetilde{R}_+(\mathbf{x}))-\frac{1}{2}M_-( \widetilde{R}_-(\mathbf{x})), \end{aligned}$$

and define $K:\mathbb {C}^N\rightarrow \mathbb {C}^N$ as:

$$\begin{aligned} K(\mathbf{x})&=2M_{\beta _{2n-5}}(J_{2n-5}(\mathbf{x}))-R_+(\mathbf{x})\circ \widetilde{R}_+(\mathbf{x})-R_-(\mathbf{x})\circ \widetilde{R}_-(\mathbf{x})\\&\quad +\sum _{s=1}^{n-3}(J_{2n-4-s}(\mathbf{x})\circ \widetilde{P}_s(\mathbf{x})+P_s(\mathbf{x})\circ \widetilde{J}_{2n-4-s}(\mathbf{x})). \end{aligned}$$

Proposition 10.3

Let $\mathfrak {g}$ be of type $D_n$, $n\ge 4$. The operator (9.1) with $X^i$ as in (10.6) has trivial monodromy at all $w_{\ell }$, $\ell =1,\ldots ,N$ for all values of $\lambda$ if and only if:

• 1. The variables $\mathbf{x}^{\beta _1},\mathbf {w} \in \mathbb {C}^N$ satisfy the system

  $$\begin{aligned}&\sum _{s=0}^{n-3}\widetilde{P}_s(\mathbf{x}^{\beta _1})\circ \widetilde{J}_{2n-4-s}(\mathbf{x}^{\beta _1})= R_+(\mathbf{x}^{\beta _1})\circ R_-(\mathbf{x}^{\beta _1}) \end{aligned}$$

  (10.10a)
  $$\begin{aligned}&2(k+2n-3)\mathbf{w}=Y_\theta (K(\mathbf{x}^{\beta _1}))+R_+(\mathbf{x}^{\beta _1}) \circ S_{\gamma _{n-2}^+}(\widetilde{R}_+(\mathbf{x}^{\beta _1}))\nonumber \\&\quad -\widetilde{R}_+(\mathbf{x}^{\beta _1})\circ S_{\beta _{n-2}^+}(R_+(\mathbf{x}^{\beta _1}))+ R_-(\mathbf{x}^{\beta _1})\circ S_{\gamma _{n-2}^-}(\widetilde{R}_-(\mathbf{x}^{\beta _1}))\nonumber \\&\quad -\widetilde{R}_-(\mathbf{x}^{\beta _1})\circ S_{\beta _{n-2}^-}(R_-(\mathbf{x}^{\beta _1})) +\frac{2}{3}k\widetilde{R}_+(\mathbf{x}^{\beta _1})\circ \widetilde{R}_-(\mathbf{x}^{\beta _1})\nonumber \\&\quad +\sum _{i=1}^{n-3}\left( J_{2n-4-i}(\mathbf{x}^{\beta _1})\circ \left( S_{\gamma _i}(\widetilde{P}_{i}(\mathbf{x}^{\beta _1}))-\frac{k}{3} P_{i}(\mathbf{x}^{\beta _1})\right) -\widetilde{J}_{2n-4-i}(\mathbf{x}^{\beta _1})\circ S_{\beta _i}(P_{i}(\mathbf{x}^{\beta _1}))\right) \nonumber \\&\quad +\sum _{i=n-1}^{2n-5}\left( P_{2n-4-i}(\mathbf{x}^{\beta _1})\circ \left( S_{\gamma _i}(\widetilde{J}_{i}(\mathbf{x}^{\beta _1}))-\frac{k}{3} J_{i}(\mathbf{x}^{\beta _1})\right) - \widetilde{P}_{2n-4-i}(\mathbf{x}^{\beta _1})\circ S_{\beta _i}(J_{i}(\mathbf{x}^{\beta _1})\right) \nonumber \\&+\frac{2}{3}k(R_+(\mathbf{x}^{\beta _1})+R_-(\mathbf{x}^{\beta _1}))\circ J_{n-1}(\mathbf{x}^{\beta _1})- \frac{2}{3}k(\widetilde{R}_+(\mathbf{x}^{\beta _1})+\widetilde{R}_-(\mathbf{x}^{\beta _1}))\circ \widetilde{J}_{n-1}(\mathbf{x}^{\beta _1})\nonumber \\&\quad -\frac{2}{3}k\sum _{i=2}^{n-3}\left( \widetilde{J}_{2n-3-i}(\mathbf{x}^{\beta _1}) \circ P_i(\mathbf{x}^{\beta _1})-J_{2n-3-i}(\mathbf{x}^{\beta _1})\circ \widetilde{P}_i(\mathbf{x}^{\beta _1})\right) , \end{aligned}$$

  (10.10b)

• 2. The variables $\mathbf{x}^\alpha \in \mathbb {C}^N$, $\alpha \in \Theta$, are given in terms of $\mathbf{x}^{\beta _1}$ as

  $$\begin{aligned}&\mathbf{x}^{\gamma _i}=\widetilde{P}_i(\mathbf{x}^{\beta _1})i=1,\ldots ,n-3,\\&\mathbf{x}^{\beta _i}=P_i(\mathbf{x}^{\beta _1}) i=1,\ldots ,n-3,\\&\mathbf{x}^{\beta ^\pm _{n-2}}=R_\pm (\mathbf{x}^{\beta _1}),\\&\mathbf{x}^{\gamma ^\pm _{n-2}}=\widetilde{R}_\pm (\mathbf{x}^{\beta _1}),\\&\mathbf{x}^{\gamma _i}=\widetilde{J}_i(\mathbf{x}^{\beta _1}) i=n-1,\ldots ,2n-5,\\&\mathbf{x}^{\beta _i}=J_i(\mathbf{x}^{\beta _1}) i=n-1,\ldots ,2n-5,\\&\mathbf{x}^\theta =K(\mathbf{x}^{\beta _1}). \end{aligned}$$

Corollary 10.4

Let $N=1$. The system (10.3) admits, for generic values of $r \in \mathfrak {h},k \in \mathbb {R}$, n solutions.

Skecth of the proof

The system (10.3) splits into an algebraic equation for the sole variable $\mathbf{{x}^{\beta _1}}=x^{\beta _1}$, and a linear equation for $\mathbf{{w}}=w_1$. By recursively computing the degree of the functions $\tilde{P}_s,\tilde{J},R_{\pm }$, one shows that (10.10a) is an equation of the form $(x-a)^{-1}\Pi _n(x^{\beta _1})=0$, where a is a complex number and $\Pi _n$ a polynomial of degree n. The coefficients of $\Pi _n$, as well as the number a, depend on r, k, so that for generic values of these parameter the equation has exactly n solutions. $\quad \square$

11.3 The case $E_6$

For $\mathfrak {g}$ of type $E_6$ we have $h^\vee =12$, $\theta =\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4+2\alpha _5+\alpha _6$, and $\dim \mathfrak {t}=22$. Since $I{\setminus } I_\theta =\{4\}$ then from (9.10) we get

$$\begin{aligned} \pi _0(f)=-E_{-\alpha _1}-E_{-\alpha _2}-E_{-\alpha _3}-E_{-\alpha _5}-E_{-\alpha _6},\quad \pi _{-1}(f)=-E_{-\alpha _4}. \end{aligned}$$

Denoting the roots:

$$\begin{aligned}&\gamma _1=\alpha _4&\beta _1=\alpha _3+\alpha _4\\&\gamma _2=\alpha _2+\alpha _3+\alpha _4&\beta _2=\alpha _1+\alpha _2+\alpha _3+\alpha _4\\&\gamma _3=\alpha _3+\alpha _4+\alpha _5&\beta _3=\alpha _2+\alpha _3+\alpha _4+\alpha _5&\\&\gamma _4=\alpha _1+\alpha _2+\alpha _3+\alpha _4+\alpha _5&\beta _4=\alpha _3+\alpha _4+\alpha _5+\alpha _6\\&\gamma _5=\alpha _2+\alpha _3+\alpha _4+\alpha _5+\alpha _6&\beta _5=\alpha _1+\alpha _2+\alpha _3+\alpha _4+\alpha _5+\alpha _6\\&\gamma _6=\alpha _2+2\alpha _3+\alpha _4+\alpha _5&\beta _6=\alpha _1+\alpha _2+2\alpha _3+\alpha _4+\alpha _5 \\&\gamma _7=\alpha _1+2\alpha _2+2\alpha _3+\alpha _4+\alpha _5&\beta _7=\alpha _2+2\alpha _3+\alpha _4+\alpha _5+\alpha _6\\&\gamma _8=\alpha _1+\alpha _2+2\alpha _3+\alpha _4+\alpha _5+\alpha _6&\beta _8=\alpha _1+2\alpha _2+2\alpha _3+\alpha _4+\alpha _5+\alpha _6 \\&\gamma _9=\alpha _2+2\alpha _3+\alpha _4+2\alpha _5+\alpha _6&\beta _9=\alpha _1+\alpha _2+2\alpha _3+\alpha _4+2\alpha _5+\alpha _6\\&\gamma _{10}=\alpha _1+2\alpha _2+2\alpha _3+\alpha _4+2\alpha _5+\alpha _6&\beta _{10}=\alpha _1+2\alpha _2+3\alpha _3+\alpha _4+2\alpha _5+\alpha _6 \end{aligned}$$

then we have $\Theta =\{0,\theta \}\cup \{\beta _i,\gamma _i: i=1,\ldots ,10\}$. Thus, (9.17) reads

$$\begin{aligned} X^i= {\left\{ \begin{array}{ll} -\mathbf{1}\otimes \theta ^\vee &{}\quad i=0,\\ \mathbf{x}^{\gamma _1}\otimes E_{\gamma _1} &{}\quad i=1,\\ \mathbf{x}^{\beta _1}\otimes E_{\beta _1} &{}\quad i=2,\\ \mathbf{x}^{\gamma _2}\otimes E_{\gamma _2}+\mathbf{x}^{\gamma _3}\otimes E_{\gamma _3} &{}\quad i=3,\\ \mathbf{x}^{\beta _2}\otimes E_{\beta _2}+\mathbf{x}^{\beta _3}\otimes E_{\beta _3}+\mathbf{x}^{\beta _4}\otimes E_{\beta _4} &{}\quad i=4,\\ \mathbf{x}^{\gamma _4}\otimes E_{\gamma _4}+\mathbf{x}^{\gamma _5}\otimes E_{\gamma _5}+\mathbf{x}^{\gamma _6}\otimes E_{\gamma _6}&{}\quad i=5,\\ \mathbf{x}^{\beta _5}\otimes E_{\beta _5}+\mathbf{x}^{\beta _6}\otimes E_{\beta _6}+\mathbf{x}^{\beta _7}\otimes E_{\beta _7}&{}\quad i=6,\\ \mathbf{x}^{\gamma _7}\otimes E_{\gamma _7}+\mathbf{x}^{\gamma _8}\otimes E_{\gamma _8}+\mathbf{x}^{\gamma _9}\otimes E_{\gamma _9} &{}\quad i=7,\\ \mathbf{x}^{\beta _8}\otimes E_{\beta _8}+\mathbf{x}^{\beta _9}\otimes E_{\beta _9} &{}\quad i=8,\\ \mathbf{x}^{\gamma _{10}}\otimes E_{\gamma _{10}} &{}\quad i=9,\\ \mathbf{x}^{\beta _{10}}\otimes E_{\beta _{10}} &{}\quad i=10,\\ \mathbf{x}^\theta \otimes E_\theta &{}\quad i=11. \end{array}\right. } \end{aligned}$$

(10.11)

and introducing the operator

$$\begin{aligned} \widetilde{M}_\alpha (\mathbf{v})=M_\alpha (\mathbf{v})-\mathbf{x}^{\gamma _1}\circ \mathbf{v}, \end{aligned}$$

for $\alpha \in \Theta$ and $\mathbf{v}\in \mathbb {C}^N$, then system (9.22) takes the form

$$\begin{aligned} \mathbf{x}^{\gamma _1}&=(k+\theta (r))\mathbf{1}-2A_0(\mathbf{1}),\\ \mathbf{x}^{\beta _1}&=\widetilde{M}_{\gamma _1}(\mathbf{x}^{\gamma _1}),\\ \mathbf{x}^{\gamma _2}-\mathbf{x}^{\gamma _3}&=\widetilde{M}_{\beta _1}(\mathbf{x}^{\beta _1}),\\ \mathbf{x}^{\beta _2}-\mathbf{x}^{\beta _3}&=\widetilde{M}_{\gamma _2}(\mathbf{x}^{\gamma _2}),\\ \mathbf{x}^{\beta _4}-\mathbf{x}^{\beta _3}&=\widetilde{M}_{\gamma _3}(\mathbf{x}^{\gamma _3}),\\ -\mathbf{x}^{\gamma _4}&=\widetilde{M}_{\beta _2}(\mathbf{x}^{\beta _2}),\\ \mathbf{x}^{\gamma _5}&=\widetilde{M}_{\beta _4}(\mathbf{x}^{\beta _4}),\\ \mathbf{x}^{\gamma _6}&=\widetilde{M}_{\beta _2}(\mathbf{x}^{\beta _2})+\widetilde{M}_{\beta _3}(\mathbf{x}^{\beta _3})+\widetilde{M}_{\beta _4}(\mathbf{x}^{\beta _4}),\\ 2\mathbf{x}^{\beta _5}&=M_{\gamma _6}(\mathbf{x}^{\gamma _6})-\widetilde{M}_{\gamma _4}(\mathbf{x}^{\gamma _4})+\widetilde{M}_{\gamma _5}(\mathbf{x}^{\gamma _5})+\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\gamma _3}-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _3},\\ 2\mathbf{x}^{\beta _6}&=M_{\gamma _6}(\mathbf{x}^{\gamma _6})+\widetilde{M}_{\gamma _4}(\mathbf{x}^{\gamma _4})+\widetilde{M}_{\gamma _5}(\mathbf{x}^{\gamma _5})+\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\gamma _3}-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _3},\\ 2\mathbf{x}^{\beta _7}&=-M_{\gamma _6}(\mathbf{x}^{\gamma _6})+\widetilde{M}_{\gamma _4}(\mathbf{x}^{\gamma _4})+\widetilde{M}_{\gamma _5}(\mathbf{x}^{\gamma _5})-\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\gamma _3}+\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _3},\\ \mathbf{x}^{\gamma _8}&=\widetilde{M}_{\beta _5}(\mathbf{x}^{\beta _5}),\\ \mathbf{x}^{\gamma _7}&=\widetilde{M}_{\beta _5}(\mathbf{x}^{\beta _5})+M_{\beta _6}(\mathbf{x}^{\beta _6})+\mathbf{x}^{\beta _2}\circ \mathbf{x}^{\gamma _3}-\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\gamma _4},\\ \mathbf{x}^{\gamma _9}&=\widetilde{M}_{\beta _5}(\mathbf{x}^{\beta _5})-M_{\beta _7}(\mathbf{x}^{\beta _7})-\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\beta _4}+\mathbf{x}^{\gamma _1}\circ \mathbf{x}^{\beta _5},\\ \mathbf{x}^{\beta _8}&=-M_{\gamma _7}(\mathbf{x}^{\gamma _7})+\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\gamma _4}-\mathbf{x}^{\beta _2}\circ \mathbf{x}^{\beta _3},\\ \mathbf{x}^{\beta _9}&=M_{\gamma _9}(\mathbf{x}^{\gamma _9})+\mathbf{x}^{\beta _3}\circ \mathbf{x}^{\beta _4}-\mathbf{x}^{\gamma _3}\circ \mathbf{x}^{\gamma _5},\\ M_{\gamma _9}(\mathbf{x}^{\gamma _9})&=\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\gamma _4}+\mathbf{x}^{\gamma _3}\circ \mathbf{x}^{\gamma _5}-\mathbf{x}^{\beta _2}\circ \mathbf{x}^{\beta _3}-\mathbf{x}^{\beta _3}\circ \mathbf{x}^{\beta _4}-\mathbf{x}^{\beta _2}\circ \mathbf{x}^{\beta _4}\\&\quad -\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _5}-M_{\gamma _7}(\mathbf{x}^{\gamma _7})-M_{\gamma _8}(\mathbf{x}^{\gamma _8}),\\ 2\mathbf{x}^{\gamma _{10}}&=M_{\beta _9}(\mathbf{x}^{\beta _9})-M_{\beta _8}(\mathbf{x}^{\beta _8})+\mathbf{x}^{\beta _4}\circ \mathbf{x}^{\gamma _4}-\mathbf{x}^{\beta _5}\circ \mathbf{x}^{\gamma _3}-\mathbf{x}^{\beta _2}\circ \mathbf{x}^{\gamma _5}+\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\beta _5},\\ M_{\beta _8}(\mathbf{x}^{\beta _8})&=-M_{\beta _9}(\mathbf{x}^{\beta _9})-\mathbf{x}^{\beta _4}\circ \mathbf{x}^{\gamma _4}+\mathbf{x}^{\beta _5}\circ \mathbf{x}^{\gamma _3}-\mathbf{x}^{\beta _2}\circ \mathbf{x}^{\gamma _5}+\mathbf{x}^{\gamma _2}\circ \mathbf{x}^{\beta _5},\\ \mathbf{x}^{\beta _{10}}&=M_{\gamma _{10}}(\mathbf{x}^{\gamma _{10}})+\mathbf{x}^{\gamma _4}\circ \mathbf{x}^{\gamma _5}-\mathbf{x}^{\beta _3}\circ \mathbf{x}^{\beta _5},\\ \mathbf{x}^{\theta }&=M_{\beta _{10}}(\mathbf{x}^{\beta _{10}})+\sum _{i=1}^5(\mathbf{x}^{\gamma _i}\circ \mathbf{x}^{\beta _{11-i}}-\mathbf{x}^{\gamma _{11-i}}\circ \mathbf{x}^{\beta _{i}}),\\ 2(k+11)\mathbf{w}&=Y_\theta (\mathbf{x}^\theta )+\sum _{i=1}^{11}\left( \mathbf{x}^{\beta _{11-i}}\circ S_{\gamma _i}(\mathbf{x}^{\gamma _i})-\mathbf{x}^{\gamma _{11-i}}\circ S_{\beta _i}(\mathbf{x}^{\beta _i})\right) \\&-\frac{2}{3}k\Big (\mathbf{x}^{\beta _1}\circ \mathbf{x}^{\beta _{10}}+\mathbf{x}^{\gamma _{10}}\circ (\mathbf{x}^{\gamma _{3}}-\mathbf{x}^{\gamma _{2}})+\mathbf{x}^{\beta _{9}}\circ (\mathbf{x}^{\beta _{2}}-\mathbf{x}^{\beta _{3}})\\&\quad +\mathbf{x}^{\beta _{8}}\circ (\mathbf{x}^{\beta _{3}}-\mathbf{x}^{\beta _{4}})+\mathbf{x}^{\gamma _{8}}\circ (\mathbf{x}^{\gamma _{5}}-\mathbf{x}^{\gamma _{4}}-\mathbf{x}^{\gamma _{6}})+\mathbf{x}^{\gamma _{9}}\circ \mathbf{x}^{\gamma _{4}}\\&\quad -\mathbf{x}^{\gamma _{7}}\circ \mathbf{x}^{\gamma _{7}}+\mathbf{x}^{\beta _{5}}\circ \mathbf{x}^{\beta _{6}}+\mathbf{x}^{\beta _{6}}\circ \mathbf{x}^{\beta _{7}}-\mathbf{x}^{\beta _{5}}\circ \mathbf{x}^{\beta _{7}}\Big ). \end{aligned}$$

12 The $\mathfrak {sl}_2$ Case

The case when $\mathfrak {g}$ is of type $A_1$, namely the Lie algebra $\mathfrak {sl}_2(\mathbb {C})$, requires a separate approach, essentially due to the fact that only in this case the spectrum of ${{\,\mathrm{ad}\,}}\theta ^\vee$ in the adjoint representation does not contain $\pm 1$, and it is thus given by

$$\begin{aligned} \sigma (\theta ^\vee )=\left\{ -2,0,2\right\} . \end{aligned}$$

(11.1)

Since this case was already considered in [8] and in [20] (see also [21]), in this section we merely show that our approach is equivalent. To this aim we work with quantum KdV opers in the canonical form (4.11), which is actually simpler in this particular case. A simple computations shows that operator (4.11) reads

$$\begin{aligned} \mathcal {L}_{\mathfrak {s}}=\partial _z+ \begin{pmatrix} 0 &{} \quad v(z)\\ 1 &{} \quad 0 \end{pmatrix}, \end{aligned}$$

(11.2)

with

$$\begin{aligned} v(z)=\frac{r_1(r_1-1)}{z^2}+\frac{1}{z}+\lambda z^k+\sum _{j=1}^N\left( \frac{2}{(z-w_j)^2}+\frac{s(j)}{z(z-w_j)}\right) . \end{aligned}$$

(11.3)

Here $r_1$ is an arbitrary complex number, $-2<k=-\hat{k}-1<-1$, and $s(j),j=1,\ldots N$ are free parameters to be determined. The equation $\mathcal {L}\psi =0$, in the first fundamental representation, can be written in the form of a second order differential equation

$$\begin{aligned} \psi ''(z)=v(z)\psi (z). \end{aligned}$$

(11.4)

It is well known that the operator (11.2) has trivial monodromy at $w_j$ if and only if the Frobenius expansion of the dominant solution $\psi _-=(z-w_j)^{-1}(1+O(z-w_j))$ of the latter equation does not contain logarithm terms. This condition imposes a polynomial relation among the first few terms of the Laurent expansion of v at $z_j$. Indeed, denoting

$$\begin{aligned} v(z)=\frac{2}{(z-w_j)^2}+\frac{a(j)}{z-w_j}+b(j)+c(j)(z-w_j)+O\big ((z-w_j)^2\big ), \end{aligned}$$

the trivial monodromy condition reads

$$\begin{aligned} -\frac{a(j)^3}{4}+a(j)b(j)-c(j)=0 \quad j=1,\ldots , N. \end{aligned}$$

(11.5)

We notice that the coefficients b(j), c(j) are affine functions of $\lambda$. Since the equation (11.5) is required to hold for any $\lambda$, then we can separate $-\frac{a(j)^3}{4}+a(j)b(j)-c(j)$ into a constant part (in $\lambda$) and a linear part (in $\lambda$), and both parts vanish identically. The linear part reads

$$\begin{aligned} a(j)w_j^k-kw^{k-1}=0, \end{aligned}$$

from which we deduce that $a(j)=k/w_j$. Since the coefficient s(j) appearing in (11.3) is related to a(j) by $a(j)=s(j)/w_j$, we thus obtain

$$\begin{aligned} s(j)=k. \end{aligned}$$

(11.6)

After some algebraic manipulations, the constant part reads

$$\begin{aligned} \widetilde{\Delta }-(k+1)w_\ell =\sum _{\begin{array}{c} j=1,\ldots ,N\\ j\ne \ell \end{array}} \frac{w_\ell ((k+2)^2w_\ell ^2-k(2k+5)w_\ell w_j+k(k+1)w_j^2)}{(w_\ell -w_j)^3}, \end{aligned}$$

(11.7)

with $\widetilde{\Delta }=\frac{k^3}{4}+k(k+1)-(k+2)r_1(r_1-1)$. The latter system provides the position of the poles, and thus, together with (11.6), fully determines the $\mathfrak {sl}_2$ quantum KdV opers (11.2).

Remark 11.1

The operator (11.2) (with $r_1=-\ell$), subject to the relations (11.6) and (11.7), was shown in [20, §5.5, §5.7] to coincide—after the change of coordinates $z=\big (\frac{k+2}{2}\big )^2 x^{\frac{2}{2+k}}$—with the operator with ‘monster potential’ originally proposed in [8, Eqs. (1), (3)].

Remark 11.2

According to the general theory developed in Sect. 6, we can write operator in the form (9.1). This reads

$$\begin{aligned} \mathcal {L}=\partial _z+ \begin{pmatrix} r_1 /z &{} \quad 1/z+\lambda z^{k}\\ 1 &{} \quad -r_1/z \end{pmatrix} +\sum _{j=1}^N \frac{1}{z-w_j} \begin{pmatrix} -1 &{} \quad x^\theta (j)/z\\ 0 &{} \quad 1 \end{pmatrix}, \end{aligned}$$

(11.8)

where $x^\theta (j)=k+2r_1-2\sum _{\ell \ne j}\frac{w_j}{w_j-w_\ell }$.

Appendix. Frobenius Solutions

Here we prove Proposition 5.10 in the case when $\hat{k}$ is irrational. We give full details in the case when the set of additional singularities $\lbrace w_j\rbrace _{j\in J}$ is empty (i.e. the ground state), omitting a few details of the general case to the reader. For convenience, we restate the proposition (in the irrational case).

Proposition 11.3

(cf. Proposition 5.10) Let $\hat{k}$ be irrational and $(r,\hat{k})$ be a generic pair. Let $\big (\omega (r-\rho ^\vee ),\chi _{\omega }\big )$ be an eigenpair composed of an eigenvalue and a corresponding eigenvector of $f-\rho ^\vee +\bar{r}$ in $L(\omega _i)$. A unique solution in $V^i(\lambda )$ is determined by the expansion

$$\begin{aligned} \chi _{\omega }(z,\lambda )=z^{- \rho ^\vee }z^{-\omega (r-\rho ^\vee )} F(z,\lambda z^{-\hat{k}} ), \end{aligned}$$

(11.9)

where $F(z,\zeta )$ is an $L(\omega _i)$-valued function which satisfies $\lim _{z \rightarrow 0} F(z,\lambda z^{-\hat{k}})=\chi _{\omega }.$ The function $F(z,\zeta )$ is analytic in z at $z=0$, and an entire function of $\zeta$, and it admits an analytic extension to an analytic function on $\big (\mathbb {C}{\setminus }\lbrace w_j\rbrace _{j\in J}\big ) \times \mathbb {C}$. In the case of the ground state, F is an entire functions of the two variables.

Proof

In order to study the equation $\mathcal {L}\psi =0$, with $\mathcal {L}$ given by (4.11), we first apply the gauge transform $z^{{{\,\mathrm{ad}\,}}\rho ^\vee }$, to get

$$\begin{aligned} \left( \partial _z+\frac{r-\rho ^\vee +f}{z}+\big (1-\lambda z^{-1+\tilde{k}} \big )e_\theta + \sum _{j\in J}\sum _{i=1}^n\sum _{l=0}^{d_i}\frac{s^{d_i}_l(j)}{(z-w_j)^{d_i+1-l}}\right) \bar{\psi }(z) =0, \end{aligned}$$

where $\bar{\psi }(z)=z^{\rho ^\vee }\psi (z)$. To simplify our notation, we write

$$\begin{aligned} \sum _{j\in J}\sum _{i=1}^n\sum _{l=0}^{d_i}\frac{s^{d_i}_l(j)}{(z-w_j)^{d_i+1-l}}=\sum _{k\ge 0}s_k z^k. \end{aligned}$$

Note that the above power series has radius of convergence $\min _{j \in J}|w_j|$. We then look for a solution of the latter equation in the form of the Frobenius-like series

$$\begin{aligned} \bar{\psi }(z)= z^{-\omega (r-\rho ^\vee )} \big ( \psi _{\omega } + \sum _{m,n \in \mathbb {N}^2{\setminus } (0,0)} c_{m,n} z^m (\lambda z^{-\hat{k}})^n\big ). \end{aligned}$$

(11.10)

Applying the operator $\mathcal {L}$ to (11.10) we obtain the recursion

$$\begin{aligned} \big ( r-\rho ^\vee +f-\omega (r-\rho ^\vee )+m-n \hat{k} \big ) c_{m,n}+ f_0 c_{m-1,n}+\sum _{k=0}^{m-1}s_k c_{m-1-k,n}- f_0 c_{m-1,n-1}=0, \end{aligned}$$

(11.11)

where $c_{0,0}:=\psi _{\omega }$ and $c_{-1,n}=c_{m,-1}=0$ for all m, n. It is readily seen that the recursion has a unique solution and that $c_{m,n}=0$ if $n>m$. The latter identity implies that if the series converges then the function

$$\begin{aligned} F(z,\zeta )= \psi _{\omega } + \sum _{m,n \in \mathbb {N}^2{\setminus } (0,0)} c_{m,n} z^m \zeta ^n \end{aligned}$$

(11.12)

satisfies $\lim _{z\rightarrow 0}F(z,z^{-\hat{k}}\lambda )=\psi _{\omega }$.

We now address the convergence of the series in the case when $s_k=0$ for every k, namely the ground state case, and omit the same discussion for the general case. We choose a norm $\Vert \cdot \Vert$ on $L(\omega _i)$ such that $\Vert \psi _{\omega }\Vert =1$ and denote by $\Vert \cdot \Vert$ also the corresponding operator norm. Because of the genericity condition on $(r,\hat{k})$, the matrix $r-\rho ^\vee +f-\omega (r-\rho ^\vee )+m+n (\tilde{k}-1)$ is invertible. Since moreover $m-n \hat{k}>\hat{k}$, for all $m,n \ne (0,0), m\ge n$, there exist $\rho ,K<\infty$ such that

$$\begin{aligned} \Vert \big (\ell -\omega (\ell )+m - n \hat{k} \big )^{-1}\Vert \le \frac{\rho }{m-n \hat{k}}, \; \Vert f_0\Vert \le \frac{K}{\rho }. \end{aligned}$$

Because of the above inequalities, we get

$$\begin{aligned} \Vert c_{m,n} \Vert \le K \frac{\Vert c_{m-1,n} \Vert + \Vert c_{m-1,n-1} \Vert }{m-n \hat{k}}, \end{aligned}$$

and due to Lemma 11.4 below, we have that

$$\begin{aligned} \sum _{m,n \ne (0,0)}\Vert c_{m,n}\Vert |z|^m |w|^n\le 1 - e^{K|z|(1+\frac{|w|}{1-\hat{k}})}. \end{aligned}$$

The latter estimate implies that $F(z,\zeta )$ given by (11.12) is an entire function of $z,\zeta$. Substituting $\zeta =\lambda z^{-\hat{k}}$, we have

$$\begin{aligned} \sum _{m,n \ne (0,0)}\Vert c_{m,n}\Vert |z|^m |\lambda z^{-\hat{k}}|^n\le 1 - e^{K(|z|+\frac{|\lambda ||z|^{1-\hat{k}}}{\hat{k}})}, \end{aligned}$$

from which it follows immediately that $\lim _{z\rightarrow 0}F(z,\lambda z^{-\hat{k}})=\psi _{\omega }$. $\quad \square$

Lemma 11.4

Let d be a real function of two integer variables $m,n \in \mathbb {N}$ that satisfies the recursion

$$\begin{aligned} d_{m,n}=K\frac{d_{m-1,n} + d_{m-1,n-1} }{m+n q}, \; q>-1, \end{aligned}$$

where, in the above formula $d_{-1,n}=0,d_{m,-1}=0$ for all m, n. Then

$$\begin{aligned} d_{m,n}=d_{0,0}\frac{K^{m}}{(1+q)^{n}(m-n)!n!}, \; \sum _{m,n \in \mathbb {N}}d_{m,n}x^my^n=d_{0,0}e^{K x(1+\frac{y}{1+q})}. \end{aligned}$$

(11.13)

In particular $d_{m,n}=0$ if $n>m$.

Proof

We make the following change of basis in the $\mathbb {Z}^2$ lattice: $m'=m-n,n'=n$, to obtain the new recursion $d'_{m',n'}=K\frac{d'_{m'-1,n'} + d'_{m',n'-1} }{m'+n' (1+q)}$. It is easily seen that the recursion has the unique solution $d'_{m',n'}=d'_{0,0}\frac{K^{m'+n'}}{(1+q)^{n'}m'!n'!}$ and $\sum _{m',n'}d'_{m',n'} x^{m'}y^{n'}=d'_{0,0}e^{x+\frac{y}{1+q}}$. The thesis follows. $\quad \square$