Feigin–Frenkel–Hernandez Opers and the $QQ-$System

Abstract

This paper represents the completion of our work on the ODE/IM correspondence for the generalised quantum Drinfeld–Sokolov models. We present a unified and general mathematical theory, encompassing all particular cases that we had already addressed, and we fill important analytic and algebraic gaps in the literature on the ODE/IM correspondence. For every affine Lie algebra ${\mathfrak {g}}$—whose Langlands dual ${\mathfrak {g}}'$ is the untwisted affinisation of a simple Lie algebra—we study a class of affine twisted parabolic Miura ${\mathfrak {g}}$-opers, introduced by Feigin, Frenkel and Hernandez. The Feigin–Frenkel–Hernandez opers 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 define the central connection matrix and Stokes matrix for these opers, and prove that the coefficients of the former satisfy the the QQ system of the quantum ${\mathfrak {g}}'$-Drinfeld–Sokolov (or quantum ${\mathfrak {g}}'$-KdV) model. If $\mathfrak {g}$ is untwisted, it is known that the trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities. We prove a surprising negative result in the case $\mathfrak {g}$ is twisted: in this case the trivial monodromy conditions have no non-trivial solutions.

1 Introduction

In this paper we study a class of affine opers, introduced by Feigin and Frenkel [25], and Frenkel and Hernandez [30], and we prove that their monodromy data provides solutions to the Bethe equations of the quantum $\mathfrak {g}'$-Drinfeld–Sokolov (or quantum $\mathfrak {g}'$-KdV) model, where $\mathfrak {g}'$ is an untwisted affine Kac-Moody algebra. This paper thus belongs to a research field called ODE/IM correspondence, which encompasses a large family of (conjectural) relations between linear differential operators and quantum integrable models [4, 5, 7, 8, 18, 20, 21, 25, 33, 43,44,45,46,47].

The FFH (Feigin–Frenkel–Hernandez) opers belongs to the class of affine twisted parabolic Miura opers, a concept that will introduce later in the paper. Here for sake of definiteness, we introduce them as concrete partial differential operator that we name FFH connections.¹ The Lie algebraic setting is as follows - see Table 4 for more details. We let $\mathfrak {g}=\,^L(\mathfrak {g}')$ be the Langlands dual Lie algebra of $\mathfrak {g}'$ so that $\mathfrak {g}$ is an affine Kac-Moody algebra of type $\tilde{\mathfrak {g}}^{(r)}$, where $\tilde{\mathfrak {g}}$ is a simply-laced simple Lie algebra and $r\in \{1,2,3\}$ is the order of a Dynkin diagram automorphism $\sigma$ of $\tilde{\mathfrak {g}}$. We denote by $\mathring{\mathfrak {g}}$ the simple Lie algebra whose Dynkin diagram is obtained from that of $\mathfrak {g}$ by removing the $0-$th node. If $r=1$ then $\mathring{\mathfrak {g}}=\tilde{\mathfrak {g}}$ while if $r>1$ then $\mathring{\mathfrak {g}}$ is the fixed-point subalgebra of $\tilde{\mathfrak {g}}$ under $\sigma$. In addition, we denote by $\mathring{\mathfrak {g}}=\mathring{\mathfrak {n}}^-\oplus \mathring{\mathfrak {h}}\oplus \mathring{\mathfrak {n}}^+$ the triangular decomposition of $\mathring{\mathfrak {g}}$, with $\mathring{\mathfrak {h}}$ a Cartan subalgebra and $\mathring{\mathfrak {n}}^+$ (resp. $\mathring{\mathfrak {n}}^-$) a positive (resp. negative) maximal nilpotent subalgebra of $\mathring{\mathfrak {g}}$.

Fixed global coordinates $(z,\lambda )$ on $\mathbb {C}^2$, the FFH connections for the algebra $\tilde{\mathfrak {g}}^{(r)}$ are, by definition, the following differential operators whose coefficients are meromorphic functions with values in $\tilde{\mathfrak {g}}$,

$$\begin{aligned} \mathcal {L}= z\,\partial _z+ k\; \lambda \, \partial _{\lambda } +&\mathring{f}+\ell + (z+\lambda )v_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +X(j) \right) . \end{aligned}$$

(1.1)

Formula (1.1) describes a family of partial differential operators, written in terms of the fixed elements $\mathring{f},v_\theta ,\theta ^\vee \in \tilde{\mathfrak {g}}$ and of the parameters $\ell , k$, $\lbrace w_j,X(j),y(j)\rbrace _{j \in J}$, where:

• $\mathring{f} \in \mathring{\mathfrak {n}}^-$ is a principal nilpotent element of $\mathring{\mathfrak {g}}$, $\theta ^\vee \in \mathring{\mathfrak {h}}$ is the coroot corresponding to the highest short root $\theta$ of $\mathring{\mathfrak {g}}$, and $v_\theta$ is a highest weight vector, of weight $\theta$, for the $\mathring{\mathfrak {g}}-$module $\tilde{\mathfrak {g}}$.

• $\ell \in \mathring{\mathfrak {h}}$ and $k \in \mathbb {R}$, $0<k<1$, are free parameters.

• J is a (possibly empty) finite-set of indexes and for every $j \in J$, $w_j \in \mathbb {C}^*$, $X(j) \in \mathring{\mathfrak {n}}^+$, and $y(j) \in \mathbb {C}$. The parameters $\lbrace w_j,X(j),y(j)\rbrace _{j \in J}$ are not free; they are constrained by the requirements that i) $w_i^r = w_j^r$ if and only if $i =j$, and ii) that (1.1) has trivial monodromy at each pole in $\mathbb {C}^*$ of the coefficients of $\mathcal {L}$, for every value of $\lambda \in \mathbb {C}$.

1.1 Description of the main results and Structure of the paper

This paper builds on our previous works [45,46,47] ([46, 47] in collaboration with D. Valeri)—where the case $J= \emptyset$ or $r=1$ was considered. A fundamental role for the results of the present work is also played by [25] by Fegin and Frenkel, where the relation between quantum $\mathfrak {g}'$-Drinfeld Sokolov systems and affine $\mathfrak {g}$-opers was suggested for the first time, as well as [30] by Frenkel and Hernandez, where it was conjectured that the appropriate $\mathfrak {g}$-opers to consider should be twisted parabolic Miura $\mathfrak {g}$-opers. In the present paper, we provide unified and general analytic, algebraic and geometric theory of FFH opers, and, as a by-product, we complete our previous works and fill important gaps in the mathematical literature. In detail:

1.1.1 Section 2. Lie algebraic preliminaries

This section contains preliminary material on the simple Lie algebras $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$, as well as on the affine algebra $\mathfrak {g}$. We also describe those $\tilde{\mathfrak {g}}$-modules and those properties of cyclic element $\mathring{f}+ v_{\theta }$, which are relevant for the deduction of the QQ system.

1.1.2 Section 3. Analytic Theory

On their face values FFH connections are abstract partial differential operators whose coefficients are meromorphic functions with value in $\tilde{\mathfrak {g}}$. There are, in principle, vast options of differential equations one may consider. In this section we describe the differential equations relevant for the ODE/IM correspondence and we study them.

Given a finite dimensional $\tilde{\mathfrak {g}}$-module V, we denote by $V(\lambda )$ the space of entire function with values in V and we let the differential operator (1.1) act as a (meromorphic) connection of the trivial bundle $\mathbb {C}^* \times V(\lambda )$. We are thus led to consider the ODE

$$\begin{aligned} \mathcal {L}\psi =0, \quad \psi : \left( \widetilde{\mathbb {C}^*} \setminus \bigcup _{j \in J} \bigcup _{l=0}^{r-1}\Pi ^{-1} (e^{\frac{2\pi i\, l}{r}} w_j)\right) \rightarrow V(\lambda ), \end{aligned}$$

(1.2)

where $\Pi :\widetilde{\mathbb {C}^*} \rightarrow \mathbb {C}^*$ is the universal covering map. We interpret the above differential equation as a first order ODE with values in the infinite-dimensional (Frechét) space of entire functions of the spectral parameter $\lambda$; in other words, the solution of (1.2) is a function of the variable z only, and we interpret $\partial _{\lambda }$ as an operator in the space of entire functions. We address the study of local and global solutions to equation (1.2) in Proposition 3.6 and Theorem 3.10. In Proposition 3.6, we prove that the the Cauchy problem for (1.2) is well-posed if the equation is restricted to any simply-connected open subset D of the domain. In Theorem 3.10, we prove a structure theorem for the space of global solutions: We notice that the space of global solutions is a module over the ring

$$\begin{aligned} \mathcal {O}'=\lbrace f: \widetilde{\mathbb {C}^*} \times \mathbb {C} \rightarrow \mathbb {C},\; f(z;\lambda )= Q(z^{-k}\lambda ), \text{ with } Q: \mathbb {C} \rightarrow \mathbb {C}\,\, \text{ entire }\rbrace . \end{aligned}$$

(1.3)

We prove that a connection of the form (1.1) has trivial monodromy about all singularities in $\widetilde{\mathbb {C}^{*}}$ if and only if the space of global solution is a free-module of rank $\dim V$ of the ring of functions $\mathcal {O}'$.²

1.1.3 Section 4, trivial monodromy equations.

In this section we study the trivial monodromy conditions for connections of the form (1.1). Following [45], we show that, fixed a basis of $\mathring{\mathfrak {n}}_+$, the trivial monodromy conditions are equivalent to a system of algebraic equations for the coefficients $\lbrace X(j),y(j),w_j\rbrace _{j \in J}$. More precisely, in Theorem 4.6 we prove that

• If $r=1$, the trivial monodromy conditions are equivalent to a complete system of $|J|(2h-2)$ equations in$|J|(2h-2)$ scalar unknowns [45];

• If $r>1$, the trivial monodromy conditions are satisfied if and only if $J=\emptyset$.

The latter result is rather surprising. We comment on it in the general discussion of the ODE/IM literature below. We add here that, in order to obviate this negative result, we compute trivial monodromy conditions for a larger class of connections which would lead in principle to solutions of the QQ system (see (4.26) and the discussion therein), still obtaining no solutions.

1.1.4 Section 5. Deduction of the QQ System

We then study $\mathcal {O}'$-bases of solutions, with distinguished asymptotic behaviour at 0 and $\infty$, in Proposition 5.3 and Theorem 5.10, respectively. In Proposition 5.3, following [45], we show, under generic non-resonant conditions on $\ell$, that there exists a basis of Frobenius solutions. These are solutions which admits the following expansion in a neighborhood of $z=0$

$$\begin{aligned} \chi _{\gamma }(z;\lambda ) = z^{-\gamma } \sum _{m\ge 0} g_m(\lambda ) z^m, \quad g_0(0)=\chi _{\gamma } , \end{aligned}$$

(1.4)

where $g_m:\mathbb {C}\rightarrow V$ is an entire function and $\chi _{\gamma }\in V$ is an eigenvector of $\mathring{f}+\ell$ with eigenvalue $\gamma$. In Theorem 5.10, using the recent result of one of the authors with Cotti and Guzzetti [16], we study the asymptotic behaviour at $\infty$ of solution to (1.2).

(1):

We prove that for each (generic) sector of amplitude $\pi h$, there exists a $\delta >0$ and basis of solutions with the asymptotic behaviour

$$\begin{aligned} \Psi _{\nu }(z;\lambda )= \left( z^{\frac{1}{h}\mathring{\rho }^\vee }\, \psi _{\nu }\right) e^{- h \nu z^{\frac{1}{h}} + C(z;\lambda ) } \left( 1+ O(z^{-\delta })\right) , \end{aligned}$$

(1.5)

as $z \rightarrow \infty$ in the given sector. Here $\psi _{\nu }$ is an eigenvector of $\mathring{f}+ v_{\theta }$ with eigenvalue $\nu$, $\mathring{\rho }^\vee$ is the dual Weyl vector, h the Coxeter number of $\mathfrak {g}$, and $C(z;\lambda )=O( z^{\frac{1}{h}-k})$ certain polynomial expression in $z^{-k}$.

(2):

We define the central connection matrix Q and the Stokes matrix $\mathcal {T}$. These are the matrix of change of basis, respectively between the basis of Frobenius solution and a basis at $\infty$ (in a sector containing the ray $\arg z=0$), and the matrix of change of basis between two consecutive bases at $\infty$. The coefficients of such matrices take values in the ring $\mathcal {O}'$, thus are entire functions. They are denoted by symbol Q and $\mathcal {T}$, because, according to the ODE/IM correspondence, they correspond to the Baxter Q operators and the Transfer Matrix of the quantum Drinfeld–Sokolov model.

(3):

We prove that if $\nu$ is a subdominant eigenvalue, namely $\nu$ is such that $e^{- h \nu z^{\frac{1}{h}}}$ goes to 0 as fast as possible along a ray of the sector, then $\Psi _{\nu }(z;\lambda )$ admits the asymptotic (1.5) in a larger sector, of amplitude at least $2 \pi h$.

We remark here that the above results fill major gaps in the literature on the ODE/IM correspondence. In fact, the existence of bases at $\infty$, outside the case $\mathfrak {sl}_2^{(1)}$, was never proven due to the lack, prior to [16], of a theory of ODEs with not meromorphic coefficients. Therefore, the matrices $Q,\mathcal {T}$ could not be defined. Moreover, the subdominant solutions was proven to exist in [45,46,47] (case $J=\emptyset$ or $r=1$) but its asymptotic was shown to hold only in a small sector, of amplitude $\pi h$; the fact that the asymptotics holds in a large sector is crucial for some applications, such as the extended QQ-system [24].

Having studied distinguished solutions at 0 and $\infty$, we apply the machinery developed in [46, 47]—following partial results in [18, 53]—to construct solutions of the Bethe Equations. First we select ${{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}$ fundamental $\tilde{\mathfrak {g}}$-modules in such a way that in each module there exists a subdominant solution along the real positive axis and that the collection of these ${{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}$ subdominant satisfy a system of non-linear relations known as $\Psi$-system (5.37). The coefficients in the expansion of the ${{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}$ subdominant solutions with respect to the basis of Frobenius solutions belong to the ring $\mathcal {O}'$—hence are entire functions—and are called generalised spectral determinants. Substituting these expansions into the $\Psi$-system, and fixing an element w of the Weyl group of $\mathring{\mathfrak {g}}$,³ one obtains a closed system of functional equations for $2{{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}$ of these entire functions, say $Q_w^{(i)}(\lambda )$, ${\widetilde{Q}}_w^{(i)}(\lambda )$, $i=1,\dots ,{{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}$. This system goes under the name of QQ-system, and it is given by:

$$\begin{aligned} \prod _{j=1}^{{{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}}\prod _{u=0}^{B_{sj}-1}Q_w^{(j)}(q^{\frac{B_{sj}-1-2u}{r}} \lambda )&=e^{\pi i D_s\langle \ell ,w({\tilde{\alpha }}_s)\rangle }Q_w^{(s)}(q^{D_s} \lambda ){\widetilde{Q}}_w^{(s)}(q^{- D_s}\lambda )\nonumber \\&\quad -e^{- \pi i D_s\langle \ell ,w({\tilde{\alpha }}_s)\rangle }Q_w^{(s)}(q^{-D_s}\lambda ){\widetilde{Q}}_w^{(s)}(q^{D_s}\lambda ) \,, \end{aligned}$$

(1.6)

for $s=1,\dots ,{{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}$. In the above formula, $q=e^{\pi i k}$, $B_{ij}=2\delta _{ij}-C_{ij}$ is the incidence matrix of $\mathring{\mathfrak {g}}$, while ${\tilde{\alpha }}_i$ are simple roots of $\tilde{\mathfrak {g}}$, and w is an arbitrary element of the Weyl group of $\tilde{\mathfrak {g}}$. As it was proven in [47], the QQ-system implies, under some genericity assumptions, the $^L\mathring{\mathfrak {g}}$ Bethe Equations:

$$\begin{aligned} \prod _{j=1}^{{{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}}e^{i\pi {\overline{C}}_{sj}\theta ^j_w}\frac{Q_w^{(j)}(q^{{\overline{C}}_{sj}}\lambda ^*)}{Q_w^{(j)}(q^{-{\overline{C}}_{sj}}\lambda ^*)}=-1,\qquad s=1,\dots ,{{\,\textrm{rank}\,}}\mathring{\mathfrak {g}}, \end{aligned}$$

(1.7)

for every zero $\lambda ^*$ of $Q^{(i)}(\lambda )$. Here ${\overline{C}}_{sj}=C_{js}D_s$ is the symmetrized Cartan matrix of $^L\mathring{\mathfrak {g}}$ and $\theta ^j_w=\langle \ell ,w({\tilde{\omega }}_j)\rangle$, with ${\tilde{\omega }}_j$ the j-th fundamental weight of $\tilde{\mathfrak {g}}$. We remark here that in other contexts, Bethe Equations different from (1.7) are associated to the algebra $^L\mathring{\mathfrak {g}}$, see e.g. [31, 54].

1.1.5 Section 6, axiomatic definition of the FFH opers

We present the axiomatic definition of FFH opers, as a special class of connection modulo the action of a Gauge group, defined by local conditions at the singular points [30]. In Theorem 6.6 we prove a normal form/rigidity theorem for FFH opers: we show that any FFH admits an essentially unique (i.e. up to the action of the Weyl group of $\mathring{\mathfrak {g}}$ on $\ell$) representative as a FFH connection (1.1).

1.1.6 Appendix A, affine twisted parabolic Miura opers.

In the last part of the paper we study the geometric structure underlying FFH connections (1.1), which is the notion of twisted parabolic Miura $\mathfrak {g}-$opers, as suggested in [30]. We develop the theory of twisted parabolic Miura $\mathfrak {g}-$opers, which were first defined in [30]. This theory builds on the theory of parabolic Miura $\mathfrak {g}-$opers, introduced (both in the finite and in the affine case) in [25] as a generalisation of the notion of Miura opers [11], as well as on the theory of twisted opers, introduced by Frenkel and Gross in [28] (in the finite dimensional case).

1.2 Brief discussion of the literature on the ODE/IM correspondence

We conclude our introduction by contextualising our results within the literature on the ODE/IM correspondence.

The research about the ODE/IM correspondence has been rapidly evolving, both in the physics and in the mathematics literature, even though convincing mathematical evidence that the correspondence actually holds was only provided recently for the simplest case Quantum KdV/Monster potentials in [13, 14].

Besides the correspondence between FFH connections and quantum Drinfeld–Sokolov models, that we study here, many more instances of the ODE/IM correspondences have been later found, including massive deformations of quantum Drinfeld–Sokolov model, the conformal limit of inhomogeneous XXZ-type chain, the sausage model, Kondo lines defects in product of chiral WZW models, see e.g. [1, 4, 5, 9, 15, 20, 24, 27, 27, 33, 35, 39, 43, 44, 46]. We notice that beyond the theoretical relevance of the ODE/IM correspondence, on the practical level the most efficient, if not the only, way to study integrable quantum field theory is to study the corresponding ODEs, see e.g. [9, 33, 39] for this point. In fact, assuming that a conjectured ODE/IM correspondence holds, the solutions of the Bethe Equations associated to an eigenstate of the model are explicitly described as spectral determinants of the associated oper, and the eigenvalues of the commuting Hamiltonians can be computed from the coefficients of the operator, as it was discovered in [8], see also [14, 25, 30, 41, 45, 52].

We conclude the introduction with a remark on the QQ-system (1.6), which, prior to [18], was known by other names in the Integrable Model literature, for example as ‘quantum Wronskian’  in [6] ($\mathfrak {sl}_2$ case) and [3] ($\mathfrak {sl}_3$ case), and as reproduction procedure in [48,49,50] (general case). The QQ-system has played a fundamental role in the ODE/IM correspondence since the seminal paper of Dorey and Tateo [21], and, after our works [46, 47], it has become the cornerstone of the ODE/IM correspondence. Inspired by our derivation of the QQ-system, Frenkel and Hernandez [30] proved that this is not a coincidence: the QQ-system holds 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 $U_q(^L\mathfrak {g})$. This discovery has in turn inspired many developments in the theory of QQ-system, which has become an object of interest in itself, see e.g [15, 24, 26, 27, 31, 32, 37, 51, 55,56,57].

2 Lie Algebra Preliminaries

Let $\mathfrak {g}'$ be an untwisted affine Kac-Moody algebra, and denote by $\mathfrak {g}=\,^L\!\mathfrak {g}'$ the (Langlands) dual Lie algebra. Then $\mathfrak {g}$ is isomorphic to an affine algebra $\tilde{\mathfrak {g}}^{(r)}$, where $\tilde{\mathfrak {g}}$ is a simple and simply-laced Lie algebra and r is the order of a Dynkin automorphism $\sigma$ of $\tilde{\mathfrak {g}}$. In this section we first review this basic construction, with the purpose of fixing the notation, and we consider some results on the representation theory of $\tilde{\mathfrak {g}}$, which is needed for the ODE/IM correspondence.

We index the nodes of the Dynkin diagram of $\mathfrak {g}$ by the set $I=\{0,1,\dots ,n\}$ as in Tables  and , and we let $C=(C_{ij})_{i,j\in I}$ be the Cartan matrix of $\mathfrak {g}$. There exists integers $(a^\vee _0,a^\vee _1,\dots ,a^\vee _n)$ and $(a_0,a_1,\dots ,a_n)$ which satisfy

$$\begin{aligned} \sum _{i\in I}a^\vee _i C_{ij}=0,\qquad \sum _{i\in I} C_{ji}a_i=0,\qquad j\in I. \end{aligned}$$

(2.1)

and are uniquely specified by setting $a_0=a^\vee _0=1$. Removing the $0-$th node from the Dynkin diagram of $\mathfrak {g}$ one gets the Dynkin diagram of a simple Lie algebra $\mathring{\mathfrak {g}}$, see Tables , , and 4. The numbers

$$\begin{aligned} h=\sum _{i\in I}a_i,\qquad h^\vee =\sum _{i\in I}a_i^\vee , \end{aligned}$$

are, respectively, the Coxeter number and the dual Coxeter number of $\mathfrak {g}$.

2.1 The simple Lie algebras $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$

Let $\tilde{\mathfrak {g}}$ one of the simply-laced simple Lie algebras as in Table 4, with Dynkin diagram as in Tables  and . Let ${\tilde{n}}$ be the rank of $\tilde{\mathfrak {g}}$ and set ${\tilde{I}}=\{1,\dots ,{\tilde{n}}\}$. Let $\{{\tilde{e}}_i,{\tilde{\alpha }}_i^\vee , {\tilde{f}}_i|i\in {\tilde{I}}\}$ be Chevalley generators of $\tilde{\mathfrak {g}}$, satisfying the relations $(i,j\in {\tilde{I}})$:

$$\begin{aligned} {[}{\tilde{\alpha }}_i^\vee ,{\tilde{\alpha }}_j^\vee ]=0,\quad [{\tilde{\alpha }}_i^\vee ,{\tilde{e}}_j]={\tilde{C}}_{ij}{\tilde{e}}_j,\quad [{\tilde{\alpha }}_i^\vee ,{\tilde{f}}_j]=-{\tilde{C}}_{ij}{\tilde{f}}_j,\quad [{\tilde{e}}_i,{\tilde{f}}_j]=\delta _{ij}{\tilde{\alpha }}_i^\vee , \end{aligned}$$

where ${\tilde{C}}=({\tilde{C}}_{ij})_{i,j\in {\tilde{I}}}$ is the Cartan matrix of $\tilde{\mathfrak {g}}$. Fix a Cartan subalgebra

$$\begin{aligned} \tilde{\mathfrak {h}}=\langle {\tilde{\alpha }}^\vee _i\,|\,i\in {\tilde{I}}\rangle , \end{aligned}$$

(2.2)

such that $\{{\tilde{\alpha }}^\vee _i | i\in {\tilde{I}}\}\subset \tilde{\mathfrak {h}}$ are simple coroots, and denote by $\{{\tilde{\alpha }}_i | i\in {\tilde{I}}\}\subset \tilde{\mathfrak {h}}^*$ the corresponding set of simple roots, such that $\langle {\tilde{\alpha }}_i^\vee ,{\tilde{\alpha }}_j\rangle ={\tilde{C}}_{ij}$. The algebra $\tilde{\mathfrak {g}}$ admits the root space decomposition

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

(2.3)

where ${\tilde{\Delta }}\subset \tilde{\mathfrak {h}}^*$ is the set of roots and $\tilde{\mathfrak {g}}_\alpha \subset \tilde{\mathfrak {g}}$ are the root spaces. Let $\tilde{\mathfrak {n}}^+$ (resp. $\tilde{\mathfrak {n}}^-$) be the nilpotent subalgebra of $\tilde{\mathfrak {g}}$ generated by $\{{\tilde{e}}_i\,|\,i\in {\tilde{I}}\}$ (resp. $\{{\tilde{f}}_i\,|\,i\in {\tilde{I}}\}$), so that

$$\begin{aligned} \tilde{\mathfrak {g}}=\tilde{\mathfrak {n}}^-\oplus \tilde{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+. \end{aligned}$$

(2.4)

Let $\sigma$ be a Dynkin diagram automorphism of $\tilde{\mathfrak {g}}$, namely a permutation of the set ${\tilde{I}}$ such that ${\tilde{C}}_{\sigma (i)\sigma (j)}={\tilde{C}}_{ij}$. Extend $\sigma$ to a Lie algebra automorphism (still denoted by $\sigma$) $\sigma \in {{\,\textrm{Aut}\,}}(\tilde{\mathfrak {g}})$ defined on Chevalley generators by ($i\in {\tilde{I}}$)

$$\begin{aligned} \sigma ({\tilde{e}}_i)={\tilde{e}}_{\sigma (i)},\quad \sigma ({\tilde{\alpha }}^\vee _i)={\tilde{\alpha }}^\vee _{\sigma (i)},\quad \sigma ({\tilde{f}}_i)={\tilde{f}}_{\sigma (i)}. \end{aligned}$$

(2.5)

The automorphism $\sigma$ induces the following gradation on $\tilde{\mathfrak {g}}$:

$$\begin{aligned} \tilde{\mathfrak {g}}=\bigoplus _{\ell =0}^{r-1}\tilde{\mathfrak {g}}_\ell ,\qquad \tilde{\mathfrak {g}}_\ell =\left\{ x\in \tilde{\mathfrak {g}}\,|\,\sigma (x)=\varepsilon ^\ell x\right\} . \end{aligned}$$

(2.6)

where

$$\begin{aligned} \varepsilon =e^\frac{2\pi i}{r} \end{aligned}$$

(2.7)

and r is the order of $\sigma$. We denote by $\mathring{\mathfrak {g}}=\tilde{\mathfrak {g}}_0$ the fixed-point subalgebra of $\tilde{\mathfrak {g}}$ under the action of $\sigma$. Then $\mathring{\mathfrak {g}}$ is a simple Lie algebra, whose Dynkin diagram is obtained from the Dynkin diagram of $\tilde{\mathfrak {g}}$ by the folding induced by $\sigma$.

Due to the numbering of the Dynkin diagrams in Table  we have that the nodes $1,\dots ,n$ lie in different orbits, so we can represent the set of orbits by $\mathring{I}=\{1\dots ,n\}\subset {\tilde{I}}$. Note, incidentally, that the Dynkin diagram of $\mathring{\mathfrak {g}}$ can also be obtained from that of the affine algebra $\mathfrak {g}$ by removing the 0-th node, so that $\mathring{I}=I\setminus \{0\}=\{1,\dots ,n\}$. For $i\in {\tilde{I}}$ we denote by $\langle i\rangle \in \mathbb {Z}^+$ the cardinality of the $i-$th $\sigma$-orbit, and we set

$$\begin{aligned} D_i=\frac{\langle i\rangle }{r},\qquad i\in {\tilde{I}}. \end{aligned}$$

(2.8)

The Cartan matrix $(C_{ij})_{i,j\in \mathring{I}}$ of $\mathring{\mathfrak {g}}$ can be obtained summing over the columns of ${\tilde{C}}$ along the orbits of $\sigma$:

$$\begin{aligned} C_{ij}=\sum _{\ell =1}^{\langle i\rangle }{\tilde{C}}_{\sigma ^\ell (i)j},\qquad i,j\in \mathring{I}. \end{aligned}$$

The elements

$$\begin{aligned} e_i=\sum _{\ell =1}^{\langle i\rangle }{\tilde{e}}_{\sigma ^\ell (i)},\quad \alpha ^\vee _i=\sum _{\ell =1}^{\langle i\rangle }{\tilde{\alpha }}^\vee _{\sigma ^\ell (i)},\quad f_i=\sum _{\ell =1}^{\langle i\rangle }{\tilde{f}}_{\sigma ^\ell (i)},\qquad i\in \mathring{I}. \end{aligned}$$

(2.9)

satisfy the relations ($i,j\in \mathring{I}$)

$$\begin{aligned} {[}\alpha ^\vee _i,\alpha ^\vee _j]=0,\quad [\alpha ^\vee _i,e_j]=C_{ij}e_j,\quad [\alpha ^\vee _i,f_j]=-C_{ij}f_j,\quad [e_i,f_j]=\delta _{ij}\alpha ^\vee _i, \end{aligned}$$

together with the Serre relations, and are therefore Chevalley generators of $\mathring{\mathfrak {g}}$. In addition, the elements

$$\begin{aligned} \alpha _i=\frac{1}{\langle i\rangle }\sum _{\ell =1}^{\langle i\rangle }{\tilde{\alpha }}_{\sigma ^\ell (i)},\qquad i\in \mathring{I} \end{aligned}$$

(2.10)

are simple roots of $\mathring{\mathfrak {g}}$, namely they satisfy $\langle \alpha _i^\vee ,\alpha _j\rangle =C_{ij}$ ($i,j\in \mathring{I}$). Denoting

$$\begin{aligned} \mathring{\mathfrak {n}}^-=\tilde{\mathfrak {n}}^-\cap \mathring{\mathfrak {g}},\qquad \mathring{\mathfrak {h}}=\tilde{\mathfrak {h}}\cap \mathring{\mathfrak {g}},\qquad \mathring{\mathfrak {n}}^+=\tilde{\mathfrak {n}}^+\cap \mathring{\mathfrak {g}}, \end{aligned}$$

(2.11)

then we obtain the triangular decomposition

$$\begin{aligned} \mathring{\mathfrak {g}}=\mathring{\mathfrak {n}}^-\oplus \mathring{\mathfrak {h}}\oplus \mathring{\mathfrak {n}}^+. \end{aligned}$$

We denote by $\mathring{\mathfrak {b}}^\pm =\mathring{\mathfrak {h}}\oplus \mathring{\mathfrak {n}}^\pm$ the corresponding Borel subalgebras. In particular, we have:

$$\begin{aligned} \mathring{\mathfrak {h}}=\langle \alpha ^\vee _i\,|\,i\in \mathring{I}\rangle =\langle \alpha ^\vee _1,\dots ,\alpha ^\vee _n\rangle , \end{aligned}$$

(2.12)

and similarly

$$\begin{aligned} \mathring{\mathfrak {h}}^*=\langle \alpha _i\,|\,i\in \mathring{I}\rangle =\langle \alpha _1,\dots ,\alpha _n\rangle . \end{aligned}$$

We introduce the following elements:

i):

Denote by $\mathring{f}\in \mathring{\mathfrak {g}}$ the element

$$\begin{aligned} \mathring{f}=\sum _{j\in \mathring{I}}f_j, \end{aligned}$$

(2.13)

which is a principal nilpotent element of $\mathring{\mathfrak {g}}$.

ii):

Denote by $\theta ^\vee \subset \mathring{\mathfrak {h}}$ and $\theta \subset \mathring{\mathfrak {h}}^*$ the elements

$$\begin{aligned} \theta ^\vee =\sum _{i\in \mathring{I}}a^\vee _i\alpha ^\vee _i,\qquad \theta =\sum _{i\in \mathring{I}}a_i\alpha ^\vee _i, \end{aligned}$$

(2.14)

where $a_i$ and $a^\vee _i$ ($i\in \mathring{I}$) are the integer coefficients satisfying (2.1). The element $\theta$ is the highest short root of $\mathring{\mathfrak {g}}$ (i.e. the highest root of $\mathring{\mathfrak {g}}$ if $r=1$); its height is $h-1$, where h is the Coxeter number of $\mathring{\mathfrak {g}}$.

iii):

Let $\{\mathring{\omega }^\vee _i | i\in \mathring{I}\}$ be fundamental coweights of $\mathring{\mathfrak {g}}$, defined by the relations $\langle \mathring{\omega }^\vee _j,\alpha _j\rangle =\delta _{ij}$ ($i,j\in \mathring{I}$). The element

$$\begin{aligned} \mathring{\rho }^\vee =\sum _{i\in \mathring{I}}\mathring{\omega }^\vee _i \end{aligned}$$

(2.15)

satisfies $\langle \mathring{\rho }^\vee ,\alpha _i\rangle =1$ ($i\in \mathring{I}$), and therefore its adjoint action induces the principal gradation on $\mathring{\mathfrak {g}}$, defined as

$$\begin{aligned} \mathring{\mathfrak {g}}=\bigoplus _{j=1-h^\vee }^{h^\vee -1}\mathring{\mathfrak {g}}^j,\qquad \mathring{\mathfrak {g}}^j=\{x\in \mathring{\mathfrak {g}}\,|\,[\mathring{\rho }^\vee ,x]=jx\}, \end{aligned}$$

(2.16)

where $h^\vee$ is the dual Coxeter number of $\mathfrak {g}$. In particular,

$$\begin{aligned} \mathring{\mathfrak {g}}^{-1}=\bigoplus _{j\in \mathring{I}}\mathbb {C}\mathring{f}_j, \end{aligned}$$

(2.17)

so that $\mathring{f}\in \mathring{\mathfrak {g}}^{-1}$.

The gradation (2.6) induced by $\sigma$ decomposes $\tilde{\mathfrak {g}}$ into a direct sum of $\mathring{\mathfrak {g}}-$modules: the subspace $\tilde{\mathfrak {g}}_0\simeq \mathring{\mathfrak {g}}$ is a $\mathring{\mathfrak {g}}-$module via the adjoint representation, if $r>1$ $\tilde{\mathfrak {g}}_{1}$ is the (unique) quasi-minuscule $\mathring{\mathfrak {g}}-$module, and if $r>2$ then $\tilde{\mathfrak {g}}_{2}$ is isomorphic to $\tilde{\mathfrak {g}}_{1}$. The weights of $\tilde{\mathfrak {g}}_1$ (and of $\tilde{\mathfrak {g}}_2$ for $r>2$) are the zero weight and the short roots of $\mathring{\mathfrak {g}}$.

Definition 2.1

Recall that the element $\theta$ given in (2.14) is the highest short root of $\mathring{\mathfrak {g}}$. We denote by $V_\theta$ the unique quasi-minuscule $\mathring{\mathfrak {g}}$-module, namely the irreducible $\mathring{\mathfrak {g}}-$module with highest weight $\theta$, and we let $v_\theta \in V_\theta$ be a highest weight vector. If $r=1$ then $V_\theta \simeq \mathring{\mathfrak {g}}\simeq \tilde{\mathfrak {g}}$ (the adjoint representation) and we identify $v_\theta$ with $e_\theta$, the highest root vector of $\tilde{\mathfrak {g}}\simeq \mathring{\mathfrak {g}}$. If $r>1$ then $V_\theta \simeq \tilde{\mathfrak {g}}_1$ (as $\mathring{\mathfrak {g}}$-modules), and in particular $v_\theta \in \tilde{\mathfrak {g}}\setminus \mathring{\mathfrak {g}}$. In both cases, we identify $V_\theta$ with the corresponding subspace of $\tilde{\mathfrak {g}}$.

The element $v_\theta$ is defined up to a nonzero scalar multiple, which will be fixed in the next section (see Theorem 2.5). Note, incidentally, that denoting by $\mathring{\mathcal {N}}$ the group

$$\begin{aligned} \mathring{\mathcal {N}}=\{\exp (y)\,|\,y\in \mathring{\mathfrak {n}}^+\}, \end{aligned}$$

(2.18)

then $v_\theta$ is fixed by $\mathring{\mathcal {N}}$, that is

$$\begin{aligned} gv_\theta =v_\theta ,\qquad g\in \mathring{\mathcal {N}}. \end{aligned}$$

(2.19)

This follows immediately from the fact that $v_\theta$ is a highest weight vector for a $\mathring{\mathfrak {g}}-$module.

Table 1 Dynkin diagram of $\tilde{\mathfrak {g}}\simeq \mathring{\mathfrak {g}}$ (case $r=1$)

Table 2 Dynkin diagram of $\tilde{\mathfrak {g}}$ with the automorphism $\sigma$ (of order $r>1$), and the Dynkin diagram of the folded algebra $\mathring{\mathfrak {g}}$. In the table, we set $D_3=A_3$

2.2 The affine Kac-Moody algebra $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$

Let $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$ be one of the affine Kac-Moody algebra listed in Table 4. We recall the loop presentation of $\mathfrak {g}$, the construction is standard [36]. Let $\mathcal {L}(\tilde{\mathfrak {g}})=\tilde{\mathfrak {g}}[\lambda ,\lambda ^{-1}]$ denote the loop algebra of $\tilde{\mathfrak {g}}$, with the natural Lie algebra structure which extends the one of $\tilde{\mathfrak {g}}$. Extend also the action of $\sigma$ from $\tilde{\mathfrak {g}}$ to $\mathcal {L}(\tilde{\mathfrak {g}})$ by letting

$$\begin{aligned}\sigma (\lambda ^m x)= \varepsilon ^{-m} \lambda ^m\sigma (x),\qquad m\in \mathbb {Z}, x\in \tilde{\mathfrak {g}},\end{aligned}$$

with $\varepsilon$ given by (2.7). Note that this action reduces to the identity when $r=1$. The affine Kac-Moody algebra $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$ is defined as

$$\begin{aligned} \mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}=\bigoplus _{k=0}^{r-1}\lambda ^k\tilde{\mathfrak {g}}_k[\lambda ^{r},\lambda ^{-r}]\oplus \mathbb {C}K\oplus \mathbb {C}{} \textbf{d}, \end{aligned}$$

(2.20)

where K is central and the element $\textbf{d}$ satisfies $[\textbf{d},\lambda ^mx]=-m\lambda ^mx$, for $m\in \mathbb {Z}$ and $x\in \tilde{\mathfrak {g}}$. Defining $\sigma (K)=K$ and $\sigma (\textbf{d})=\textbf{d}$ then $\tilde{\mathfrak {g}}^{(r)}$ is $\sigma$-invariant. Let $I=\{0,1,\dots ,n\}=\{0\}\cup \mathring{I}$. The Chevalley generators $\{e_i,\alpha ^\vee _i,f_i \, |\, i\in I\}$ of $\tilde{\mathfrak {g}}^{(r)}$ can be written in terms of $\tilde{\mathfrak {g}}$ as follows. The elements $\{e_i,\alpha ^\vee _i,f_i \, |\, i\in \mathring{I}\}$ are given by (2.9), in particular, they generate the simple Lie algebra $\tilde{\mathfrak {g}}_0\simeq \mathring{\mathfrak {g}}$ described above. On the other hand, the generators $e_0$, $\alpha ^\vee _0$ and $f_0$ are given by

$$\begin{aligned} f_0=\lambda v_\theta ,\qquad \alpha ^\vee _0=K-\theta ^\vee ,\qquad e_0=\lambda ^{-1}v_{-\theta }, \end{aligned}$$

(2.21)

where $\theta ^\vee \in \mathring{\mathfrak {h}}$ is given by (2.14), the element $v_{\theta }\in \tilde{\mathfrak {g}}$ was introduced in Definition 2.1, and $v_{-\theta }\in \tilde{\mathfrak {g}}$ is uniquely defined by the relation $(v_\theta |v_{-\theta })=1$, where $(\cdot |\cdot )$ is the normalized invariant bilinear form on $\tilde{\mathfrak {g}}$. Finally, the scaling element $\textbf{d}$ is realized as

$$\begin{aligned} \textbf{d}=-\lambda \partial _\lambda . \end{aligned}$$

(2.22)

Remark 2.2

The choice of the negative sign in (2.22) together with (2.21) implies the relations $[\textbf{d},e_0]=e_0$, $[\textbf{d},f_0]=-f_0$.

We define

$$\begin{aligned} f=\sum _{i\in I}f_i=\sum _{i\in \mathring{I}}f_i+f_0=\mathring{f}+f_0, \end{aligned}$$

(2.23)

where $\mathring{f}$ is the principal nilpotent element (2.13). Equivalently, we can write

$$\begin{aligned} f=\mathring{f}+\lambda v_\theta . \end{aligned}$$

(2.24)

We extend the action of the group $\mathring{\mathcal {N}}$ to $\mathfrak {g}$ by setting

$$\begin{aligned} g(\lambda ^mx)=\lambda ^m gx,\qquad gK=K,\qquad g\textbf{d}=\textbf{d}, \end{aligned}$$

(2.25)

for $g\in \mathring{\mathcal {N}}$, $m\in \mathbb {Z}$, $x\in \tilde{\mathfrak {g}}$. In particular, due to (2.19) we have

$$\begin{aligned} gf_0=g(\lambda v_\theta )=\lambda gv_\theta =\lambda v_\theta =f_0. \end{aligned}$$

(2.26)

2.3 $\tilde{\mathfrak {g}}$-modules

Let $\tilde{\mathfrak {g}}$ be the simple Lie algebra (2.3) with Cartan subalgebra (2.2). Denote by $P(\tilde{\mathfrak {g}})\subset \tilde{\mathfrak {h}}^*$ the weight lattice of $\tilde{\mathfrak {g}}$ and by $P^+(\tilde{\mathfrak {g}})$ the set of dominant integral weights. For every $\omega \in P^+(\tilde{\mathfrak {g}})$ we denote by $L(\omega )$ the irreducible highest weight $\tilde{\mathfrak {g}}$-module with highest weight $\omega$, and we let $P_\omega (\tilde{\mathfrak {g}})\subset P(\tilde{\mathfrak {g}})$ be the set of weights of $L(\omega )$. Let $\{{\tilde{\omega }}_i\,|\,i\in {\tilde{I}}\}\subset P^+(\tilde{\mathfrak {g}})$ be fundamental weights of $\tilde{\mathfrak {g}}$, satisfying $\langle {\bar{\alpha }}^\vee _i,{\tilde{\omega }}_j\rangle =\delta _{ij}$ ($i,j\in {\tilde{I}}$). The corresponding $\tilde{\mathfrak {g}}$-modules $L({\tilde{\omega }}_i)$ ($i\in {\tilde{I}}$) are known as fundamental $\tilde{\mathfrak {g}}$-modules. For every $i\in {\tilde{I}}$ we fix a highest weight vector $v_i$ of $L({\tilde{\omega }}_i)$. Let ${\widetilde{B}}=({\widetilde{B}}_{ij})_{i,j\in {\tilde{I}}}$ be the incidence matrix of the Dynkin diagram of $\tilde{\mathfrak {g}}$, namely ${\widetilde{B}}_{ij}=2\delta _{ij}-{\widetilde{C}}_{ij}$ ($i,j\in {\tilde{I}}$). Note that ${\widetilde{B}}_{ij}\ge 0$. Define the dominant weights

$$\begin{aligned} \eta _i=\sum _{j\in {\tilde{I}}}{\widetilde{B}}_{ij}{\tilde{\omega }}_j\qquad i\in {\tilde{I}}. \end{aligned}$$

Then as proved in [46] for every $i\in {\tilde{I}}$ the weight $\eta _i$ is a highest weight (of multiplicity one) of the following two $\tilde{\mathfrak {g}}-$modules:

$$\begin{aligned} \bigwedge ^2 L({\tilde{\omega }}_i),\qquad \bigotimes _{j\in {\tilde{I}}}L({\tilde{\omega }}_j)^{\otimes {\widetilde{B}}_{ij}}, \end{aligned}$$

with highest weight vector given, respectively, by ${\bar{f}}_iv_i\wedge v_i$ and $\otimes _{j\in {\tilde{I}}}v_j^{{\widetilde{B}}_{ij}}$ ($i\in {\tilde{I}}$). By complete reducibility, for every $i\in {\tilde{I}}$ there exists a $\tilde{\mathfrak {g}}-$module $U_i$ such that $\bigwedge ^2 L({\tilde{\omega }}_i)=L(\eta _i)\oplus U_i$, and an morphism of representations

$$\begin{aligned} {\widetilde{m}}_i:\bigwedge ^2 L({\tilde{\omega }}_i)\rightarrow \bigotimes _{j\in \mathring{I}}\bigotimes _{\ell =0}^{\langle j\rangle -1}L({\tilde{\omega }}_{\sigma ^\ell (j)})^{\otimes {\widetilde{B}}_{i\sigma ^\ell (j)}} \end{aligned}$$

(2.27)

uniquely fixed by the conditions ${{\,\textrm{Ker}\,}}{\widetilde{m}}_i=U_i$ and ${\widetilde{m}}_i({\bar{f}}_iv_i\wedge v_i)=\otimes _{j\in {\tilde{I}}}v_j^{{\widetilde{B}}_{ij}}.$

We now consider twisted $\tilde{\mathfrak {g}}-$modules, and we extend the action of $\sigma$ on $\tilde{\mathfrak {g}}-$modules. If V is a $\tilde{\mathfrak {g}}$-module and $\Phi :\tilde{\mathfrak {g}}\rightarrow {{\,\textrm{End}\,}}(V)$ the corresponding representation, then we define the twisted representation as

$$\begin{aligned} \Phi ^\sigma =\pi \circ \sigma ^{-1}:\tilde{\mathfrak {g}}\rightarrow {{\,\textrm{End}\,}}(V), \end{aligned}$$

(2.28)

and we denote by $V^\sigma$ the vector space V with the $\tilde{\mathfrak {g}}-$module structure induced by $\Phi ^\sigma$. We define the action of $\sigma$ on $\tilde{\mathfrak {h}}^*$ by

$$\begin{aligned} \langle h,\sigma \omega \rangle =\langle \sigma ^{-1}h,\omega \rangle , \qquad h\in \tilde{\mathfrak {h}},\omega \in \tilde{\mathfrak {h}}^*, \end{aligned}$$

so that in particular $\sigma {\tilde{\omega }}_i={\tilde{\omega }}_{\sigma (i)}$ for every fundamental weight ${\tilde{\omega }}_i$ ($i\in {\tilde{I}}$). The following lemma is an elementary extension of a result proved in [47]:

Lemma 2.3

Let $L(\omega )$ be an irreducible $\tilde{\mathfrak {g}}-$module with highest weight $\omega$. Then $L(\omega )^\sigma$ is irreducible and there exists an isomorphism of $\tilde{\mathfrak {g}}-$modules $L(\omega )^\sigma \simeq L(\sigma \omega ).$ In particular, for every $i\in {\tilde{I}}$ the $\tilde{\mathfrak {g}}-$module $L({\tilde{\omega }}_i)^\sigma$ is isomorphic to the fundamental $\tilde{\mathfrak {g}}-$module $L({\tilde{\omega }}_{\sigma (i)})$.

Proof

It is easy to show that that a finite dimensional $\tilde{\mathfrak {g}}-$module V is irreducible if and only if $V^\sigma$ is irreducible. If $v\in L(\omega )$ is an highest weight vector, namely $\Phi ({\bar{e}}_i)v=0$ and $\Phi ({\bar{\alpha }}^\vee _i)v=\langle {\bar{\alpha }}^\vee _i,\omega \rangle v$, then $\Phi ^\sigma ({\bar{e}}_i)v=\Phi (\sigma ^{-1}{\bar{e}}_i)v=\Phi ({\bar{e}}_{\sigma ^{-1}(i)})v=0$ and $\Phi ^\sigma ({\bar{\alpha }}^\vee _i)v=\Phi (\sigma ^{-1}{\bar{\alpha }}^\vee _i)v=\langle \sigma ^{-1}{\bar{\alpha }}^\vee _i,\omega \rangle v=\langle {\bar{\alpha }}^\vee _i,\sigma \omega \rangle v$. $\square$

For every finite dimensional $\tilde{\mathfrak {g}}-$module V we define a $\mathbb {C}-$linear bijective map, denoted $\sigma$ by abuse of notation

$$\begin{aligned} \sigma :V\rightarrow V^\sigma ,\qquad v\mapsto \sigma (v), \end{aligned}$$

(2.29)

satisfying

$$\begin{aligned} \sigma (\Phi (x)v)=\Phi ^\sigma (\sigma x)\sigma (v),\qquad x\in \tilde{\mathfrak {g}}, v\in V. \end{aligned}$$

(2.30)

In particular, (2.29) maps highest weight vectors to highest weight vectors. In the case when $V=L({\tilde{\omega }}_i)$ is the $i-$th fundamental $\tilde{\mathfrak {g}}-$module, using Lemma 2.3 we thus obtain, for each $i\in {\tilde{I}}$, a $\mathbb {C}-$linear map

$$\begin{aligned} \sigma :L({\tilde{\omega }}_i)\rightarrow L({\tilde{\omega }}_{\sigma (i)}), \end{aligned}$$

(2.31)

uniquely specified, up to a nonzero scalar multiple, by condition (2.30). If $v_i\in L({\tilde{\omega }}_i)$ ($i\in {\tilde{I}}$) is a highest weight vector, then we can choose a normalization so that (2.30) reads

$$\begin{aligned} \sigma (\Phi _i(x)v_i)=\Phi _{\sigma (i)}(\sigma x)v_{\sigma (i)},\qquad x\in \tilde{\mathfrak {g}}. \end{aligned}$$

(2.32)

Furthermore, if $\sigma (i)=i$ then the map $v\mapsto \sigma (v)$ is an automorphism of $L({\tilde{\omega }}_i)$, and (2.32) reduces to

$$\begin{aligned} \sigma (\Phi _i(x)v_i)=\Phi _{i}(\sigma x)v_{i},\qquad x\in \tilde{\mathfrak {g}}. \end{aligned}$$

For every $i\in {\tilde{I}}$ we finally introduce the $\mathbb {C}-$linear map $R_i:L({\tilde{\omega }}_i)\rightarrow L({\tilde{\omega }}_i)$ as

$$\begin{aligned} R_i(v)= {\left\{ \begin{array}{ll} v &{} i\ne \sigma (i),\\ \sigma (v) &{}i=\sigma (i). \end{array}\right. } \end{aligned}$$

(2.33)

These maps will be useful later to construct the so called $\Psi -$system for $\mathfrak {g}$.

2.4 Cyclic elements of $\tilde{\mathfrak {g}}$

Given the set ${\tilde{I}}$ of the vertices of the Dynkin diagram of $\tilde{\mathfrak {g}}$ we introduce a bipartition [12] of the form ${\tilde{I}}={\tilde{I}}_1\cup {\tilde{I}}_2$ such that $1\in {\tilde{I}}_1$ and all edges of the Dynkin diagram of $\tilde{\mathfrak {g}}$ lead from ${\tilde{I}}_1$ to ${\tilde{I}}_2$. Then, we define the function $p:{\tilde{I}}\longrightarrow \mathbb {Z}/2\mathbb {Z}$ as

$$\begin{aligned}p(i)= {\left\{ \begin{array}{ll} 0\quad i\in {\tilde{I}}_1,\\ 1\quad i\in {\tilde{I}}_2. \end{array}\right. } \end{aligned}$$

Note that $p(\sigma (i))=p(i)$ ($i\in {\tilde{I}}$).

Table 3 The values of the scalars $\kappa _i$. Note that $\kappa _{\sigma (i)}=\kappa _i$ ($i\in {\tilde{I}}$)

For $t\in \mathbb {R}$ introduce the following cyclic elements of $\tilde{\mathfrak {g}}$

$$\begin{aligned} \Lambda (t)=\mathring{f}+ e^{2 \pi it}v_\theta , \end{aligned}$$

(2.34)

where $\mathring{f}$ is given in (2.13) and $v_\theta \in \tilde{\mathfrak {g}}$ is given in Definition 2.1.

Definition 2.4

Let A be an endomorphism of a vector space V. We say that a eigenvalue $\mu$ of A is maximal if it is real, its algebraic multiplicity is one, and $\mu > {\text {Re}}\mu '$ for every eigenvalue $\mu '\ne \mu$ of A.

Recall that $v_\theta \in \tilde{\mathfrak {g}}$ is defined up to a nonzero scalar multiple.

Theorem 2.5

[47, 4.7] Let $\Lambda (t)$ be given by (2.34) and let $\kappa _i$ ($i\in {\tilde{I}}$) be defined as in Table 3. We can always choose the element $v_\theta$ such that for all $i\in \mathring{I}$, the following facts hold true:

(i):

For $\ell =0,\dots ,\langle i\rangle -1$, the matrix representing the element

$$\begin{aligned} \Lambda (\kappa _i-\tfrac{\ell }{r}) \end{aligned}$$

in the fundamental $\tilde{\mathfrak {g}}-$module $L(\omega _{\sigma ^\ell (i)})$ has a maximal eigenvalue $\mu ^{(\sigma ^\ell (i))}$ and, in particular, $\mu ^{(1)}=1$. We denote by $\psi ^{(\sigma ^\ell (i))}$ the corresponding unique (up to a constant factor) eigenvector. Moreover,

$$\begin{aligned} \mu ^{(\sigma ^\ell (i))}=\mu ^{(i)},\qquad \psi ^{(\sigma ^\ell (i))}=\sigma ^\ell \left( \psi ^{(i)}\right) , \end{aligned}$$

where the action of $\sigma$ on $L({\tilde{\omega }}_i)$ is defined in (2.31).

(ii):

given $D_i$ as in (2.8), the matrix representing the element

$$\begin{aligned} \Lambda (\kappa _i-\tfrac{D_i}{2}),\qquad i\in \mathring{I}. \end{aligned}$$

(2.35)

in the $\tilde{\mathfrak {g}}-$module $\bigwedge ^2\,L({\tilde{\omega }}_i)$ has a maximal eigenvalue

$$\begin{aligned} \left( e^{-\frac{\pi \sqrt{-1}D_i}{h}}+e^{\frac{\pi \sqrt{-1}D_i}{h}}\right) \mu ^{(i)}. \end{aligned}$$

(2.36)

The corresponding eigenvector is given by

$$\begin{aligned} \psi ^{(i)}_\wedge = R_i\left( e^{\frac{\pi \sqrt{-1} D_i}{h}\mathring{\rho }^\vee }\psi ^{(i)}\right) \wedge e^{-\frac{\pi \sqrt{-1} D_i}{h}\mathring{\rho }^\vee }\psi ^{(i)}, \end{aligned}$$

(2.37)

where $R_i$ is given by (2.33).

(iii):

for the matrix representing the element (2.35) in the $\tilde{\mathfrak {g}}-$module $\bigotimes _{j\in {\tilde{I}}}L({\tilde{\omega }}_j)^{\otimes {\bar{B}}_{ij}}$ the scalar (2.36) is maximal eigenvalue. The corresponding eigenvector is

$$\begin{aligned} \psi ^{(i)}_\otimes =\bigotimes _{j\in \mathring{I}}\bigotimes _{\ell =0}^{\langle j\rangle -1}\left( e^{\frac{\pi \sqrt{-1} (\kappa _j-\kappa _i+D_i/2-\ell /r)}{h}\mathring{\rho }^\vee }\psi ^{(\sigma ^\ell (j))}\right) ^{\otimes {\widetilde{B}}_{i\sigma ^{\ell }(j)}} \end{aligned}$$

(2.38)

(iv):

We can normalize the maps ${\widetilde{m}}_i$ ($i\in \mathring{I}$) given in (2.27) such that the following identity, known as algebraic $\Psi -$system, holds:

$$\begin{aligned} {\widetilde{m}}_i(\psi ^{(i)}_\wedge )=\psi ^{(i)}_\otimes \end{aligned}$$

(2.39)

From now on, we will fix the element $v_\theta$ such that the above theorem holds.

Table 4 Simple Lie algebras and affine Kac-Moody algebras

2.5 Simplifications in the case r=1

For the benefit of the reader we specialise the thesis of Theorem 2.5, namely the algebraic $\Psi$-system, to the untwisted case $r=1$. In this case the algebras $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$ coincide, as well as the index sets $\mathring{I}={\tilde{I}}=\{1,\dots ,n\}$, and the theory greatly simplifies. The homomorphisms of $\tilde{\mathfrak {g}}$-modules defined in (2.27) reads

$$\begin{aligned} {\widetilde{m}}_i:\bigwedge ^2 L({\tilde{\omega }}_i)\rightarrow \bigotimes _{j\in {\tilde{I}}} L(\tilde{\omega _j})^{\otimes {\widetilde{B}}_{ij}}, \qquad I \in {\tilde{I}}. \end{aligned}$$

(2.40)

Theorem 2.5 reduces to the following statements. For every $i \in {\tilde{I}}$,

i):

The matrix representing $\Lambda (\frac{p(i)}{2})$ in the fundamental module $L(\omega _i)$ has a unique maximal eigenvalue, $\mu ^{(i)}$; a corresponding eigenvector is denoted by $\psi ^{(i)}$.

ii):

The matrix representing $\Lambda (\frac{1-p(i)}{2})$ in the module $\bigwedge ^2 L({\tilde{\omega }}_i)$ has a maximal eigenvalue $\left( e^{-\frac{\pi \sqrt{-1}}{h}}+e^{\frac{\pi \sqrt{-1}}{h}}\right) \mu ^{(i)}$, with eigenvector

$$\begin{aligned}&\psi ^{(i)}_\wedge =e^{\frac{\pi \sqrt{-1} }{h}\mathring{\rho }^\vee }\psi ^{(i)}\wedge e^{-\frac{\pi \sqrt{-1} }{h}\mathring{\rho }^\vee }\psi ^{(i)}. \end{aligned}$$

(2.41)

iii):

The matrix representing $\Lambda (\frac{1-p(i)}{2})$ in the module $\bigotimes _{j\in \mathring{I}} L({\tilde{\omega }}_j)^{\otimes {\widetilde{B}}_{i}}$ has a maximal eigenvalue $\left( e^{-\frac{\pi \sqrt{-1}}{h}}+e^{\frac{\pi \sqrt{-1}}{h}}\right) \mu ^{(i)}$, with eigenvector

$$\begin{aligned}&\psi ^{(i)}_\otimes =\bigotimes _{j\in {\tilde{I}}}\left( \psi ^{(j)}\right) ^{\otimes {\widetilde{B}}_{ij}}. \end{aligned}$$

(2.42)

iv):

We can normalize the maps ${\widetilde{m}}_i$ ($i\in {\tilde{I}}$) in such a way that ${\widetilde{m}}_i(\psi ^{(i)}_\wedge )=\psi ^{(i)}_\otimes$.

3 Solutions Space

In this Section, fixed an affine Kac-Moody algebra $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$ as in Table 4 and a FFH connection $\mathcal {L}$ of the form (1.1), we define local and global space of solutions for the differential equation $\mathcal {L}\psi =0$ and we provide a precise definition to the notion of trivial monodromy about the additional singular points.

These results serve as preparation to the analysis of the trivial monodromy conditions, Sect. 4, and to the construction of solutions to the QQ system as generalised spectral determinants of FFH opers, Sect. 5.

3.1 Feigin–Frenkel–Hernandez connections

Let z be a local coordinate over the punctured complex plane $\mathbb {C}^*$. Let $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$ be an affine Kac-Moody algebra as listed in Table 4. Recall the elements $f,\textbf{d}\in \mathfrak {g}$ as well as $v_\theta \in \tilde{\mathfrak {g}}$ introduced in Definition 2.1. We consider the following family of meromorphic connections with values in $\tilde{\mathfrak {g}}^{(1)}$:

$$\begin{aligned} \mathcal {L}(z)=&\partial _z+\frac{1}{z}\left( f+\ell -k \textbf{d}+ zv_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +X(j) \right) \right) . \end{aligned}$$

(3.1)

Here J is a possibly empty finite set, and the parameters $\ell , k$, and $(w_j,X(j),y(j))$, $j\in J$ satisfy the following requirements

• $\ell \in \mathring{\mathfrak {h}}$;

• $0<k<1$;

• $w_j \in {\mathbb {C}}^*, j \in J$ and $\big (w_j/w_i\big )^r \ne 1$ if $i \ne j$;

• $X(j) \in \mathring{\mathfrak {n}}^+,\,\, j \in J$.

The FFH connections will be later defined as the subclass of connections (3.1) with trivial monodromy at each $w_j$ $(j\in J)$, see Definition 3.7 below. We remark that since $\textbf{d}= -\lambda \partial _\lambda$, (3.1) can be also tought of as a first-order linear partial differential operator whose coefficients are meromorphic functions with values in $\tilde{\mathfrak {g}}$, namely

$$\begin{aligned} z \mathcal {L}= z\,\partial _z+ k\; \lambda \, \partial _{\lambda } +&\mathring{f}+\ell + (z+\lambda )v_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +X(j) \right) . \end{aligned}$$

Remark 3.1

Let $\widetilde{\mathbb {C}^*}$ be the universal cover of $\mathbb {C}^*=\mathbb {C}\setminus \{0\}$ and $\Pi :\,\widetilde{\mathbb {C}^*}\rightarrow \mathbb {C}^*$ be the corresponding projection. The operator (3.1), which is defined on the complex plane with a cut, naturally extends to a linear differential operator on

$$\begin{aligned} \widetilde{\mathbb {C}^*_J}=\left( \widetilde{\mathbb {C}^*} \setminus \bigcup _{j \in J} \bigcup _{l=0}^{r-1}\Pi ^{-1} (\varepsilon ^l w_j)\right) , \end{aligned}$$

(3.2)

given explicitly by

$$\begin{aligned} \mathcal {L}(z)=&\partial _{\Pi (z)}+\frac{1}{\Pi (z)}\left( f+\ell -k \textbf{d}+ \Pi (z)v_\theta + \sum _{j \in J} \frac{ r \Pi (z^{r})}{\Pi (z^r)-w_j^r}\left( -\theta ^\vee +X(j) \right) \right) . \end{aligned}$$

(3.3)

By abuse of notation, we omit from now on the projection $\Pi$, writing for instance (3.1) in place of (3.3). Since we have assumed that the monodromy about each additional pole is trivial, for any fixed $\lambda$ every local solution to equation (3.1) can be analytically continued to a global single-valued analytic function on $\widetilde{\mathbb {C}^*_J}$. In what follows we will be interested in solutions of (3.1) which are analytic in x and $\lambda$ separately in the domain $\widetilde{\mathbb {C}^*_J} \times \widetilde{\mathbb {C}^*}$.

3.1.1 Rotated and twisted connections.

Given $t\in \mathbb {R}$, the rotated connection is the connection induced by the map $z\mapsto e^{2\pi it}z$, that is:

$$\begin{aligned} \mathcal {L}_t(z)= \partial _z&+\frac{1}{z}\Bigg (f+\ell -k \textbf{d}+ e^{2\pi it}zv_\theta + \sum _{j \in J} \frac{ re^{2\pi irt} z^{r}}{e^{2\pi irt}z^r-w_j^r}\left( \alpha _0^\vee + X(j) \right) \Bigg ). \end{aligned}$$

(3.4)

On the other hand, the twisted connection is the operator obtain from (3.1) by acting with the automorphism $\sigma$. Since all terms in (3.1) are fixed by $\sigma$ except for $v_\theta$ (which satisfies $\sigma (v_\theta )=\varepsilon v_\theta$), the twisted operator is the operator

$$\begin{aligned} \mathcal {L}^\sigma (z)= \partial _z+\frac{1}{z}\left( f+\ell -k \textbf{d}+\varepsilon zv_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( \alpha _0^\vee +X(j) \right) \right) , \end{aligned}$$

(3.5)

where $\varepsilon =e^{\frac{2\pi i}{r}}$. A direct computation then shows that the connections of the form (3.1) satisfy the identity

$$\begin{aligned} \mathcal {L}^\sigma _{-\frac{1}{r}}(z)=\mathcal {L}(z), \end{aligned}$$

(3.6)

which is crucial in the deduction of the Bethe Equations.

3.1.2 Loop realization

It is often useful to consider a different realization of the connection (3.1), taking values in the loop algebra $\tilde{\mathfrak {g}}[\lambda ,\lambda ^{-1}]$ (or, more precisely, in the current algebra $\tilde{\mathfrak {g}}[\lambda ]$). Given $\mathcal {L}(z)$ as in (3.1), we then define

$$\begin{aligned} \mathcal {L}(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f}+\ell +(z+z^k\lambda )v_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +X(j) \right) \right) . \end{aligned}$$

(3.7)

and we call it the loop realization of (3.1). At least formally, $\mathcal {L}(z;\lambda )$ is obtained from $\mathcal {L}$ by the action of the Gauge $z^{k\textbf{d}}$, namely

$$\begin{aligned} \mathcal {L}(z;\lambda )=z^{k\textbf{d}}\mathcal {L}(z). \end{aligned}$$

The precise analytic and algebraic meaning of this Gauge will be elucidated, respectively, in Lemma 3.12 and in (A.42). While the connection $\mathcal {L}$ is a meromorphic connection over $\mathbb {C}$, its loop realization is not, since $z=0$ is a branch point of its coefficients. Given $\mathcal {L}(z;\lambda )$ as in (3.7) and $t\in \mathbb {R}$, we define the rotated connection $\mathcal {L}_t(z;\lambda )$ as

$$\begin{aligned} \mathcal {L}_t(z;\lambda )= \partial _z+\frac{1}{z}&\Bigg (\mathring{f}+\ell + (e^{2\pi it}z+z^k\lambda )v_\theta + \sum _{j \in J} \frac{ re^{2\pi irt} z^{r}}{e^{2\pi irt}z^r-w_j^r}\left( -\theta ^\vee + X(j) \right) \Bigg ). \end{aligned}$$

(3.8)

Note that the map $\mathcal {L}(z;\lambda ) \mapsto \mathcal {L}_t(z;\lambda )$ is not induced by the map $z\mapsto e^{2\pi it}z$, but rather by

$$\begin{aligned} (z,\lambda )\mapsto (e^{2\pi it}z,e^{-2\pi ikt}\lambda ),\qquad t\in \mathbb {R}. \end{aligned}$$

(3.9)

3.1.3 Dorey-Tateo symmetry

Since the connection $\mathcal {L}$ (3.1) is meromorphic at $z=0$ then $\mathcal {L}_1(z)=\mathcal {L}(z)$. This is equivalent to the following identity for the loop realisation of $\mathcal {L}$,

$$\begin{aligned} \mathcal {L}_1(z;\lambda )=\mathcal {L}(z;\lambda ). \end{aligned}$$

(3.10)

In the context of the ODE/IM correspondence, the identity (3.10) is known as Dorey-Tateo symmetry or Symanzik rescaling. We have just shown that the Dorey-Tateo symmetry is nothing but the ‘loop counterpart’  of the fact that the connection (3.1) is meromorphic at $z=0$.

Remark 3.2

In what follows, we will import results from our previous papers [45,46,47], where we studied the loop realization (3.7) of connections of the form (3.1) respectively in the cases $r=1$ and $J=\emptyset$ (simply-laced, ground state), $r>1$ and $J=\emptyset$ (non simply-laced, ground state), and $r=1$ and $J \ne \emptyset$ (simply laced, higher states). We note here that in [46, 47] we used a different coordinate system on $\widetilde{{\mathbb {C}}^*}$. In fact, while in the case $J = \emptyset$, the connection (3.7) reads

$$\begin{aligned} \mathcal {L}(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f}+\ell +\left( z+z^{k}\lambda \right) v_\theta \right) , \end{aligned}$$

(3.11)

in [46, 47], we considered connections of the form

$$\begin{aligned} L(x;E)=\partial _x+\mathring{f}+\frac{l}{x}+(x^{Mh}-E)v_\theta , \end{aligned}$$

(3.12)

with $l\in \mathring{\mathfrak {h}}$, $M>0$, and E a complex parameter. With the following change of coordinates and parameters [25]

$$\begin{aligned}&x=\varphi (z)=(h(M+1))^{\frac{1}{M+1}}z^{\frac{1}{h(M+1)}}, \\&\ell =\frac{1}{h(M+1)}(l+\mathring{\rho }^\vee ), \; k=\frac{1}{M+1}, \;\lambda =-\left( \frac{1}{h(M+1)}\right) ^{\frac{hM}{M+1}}E, \end{aligned}$$

one readily verifies that

$$\begin{aligned} \mathcal {L}(z;\lambda )=z^{{{\,\textrm{ad}\,}}\mathring{\mathfrak {\rho }}^\vee }\varphi '(z)^{{{\,\textrm{ad}\,}}\mathring{\mathfrak {\rho }}^\vee }(\varphi ^*L)(z;E). \end{aligned}$$

3.2 The space of global solutions

For every finite dimensional $\widetilde{\mathfrak {g}}$ module V we define

$$\begin{aligned} V(\lambda )=V\otimes \mathcal {O}_{\lambda }, \end{aligned}$$

(3.13)

where $\mathcal {O}_\lambda$ is the ring of entire functions in the variable $\lambda$. We let the differential operator (3.1) act as a (meromorphic) connection of the trivial bundle $\mathbb {C}^* \times V(\lambda )$. We are thus led to consider the ODE

$$\begin{aligned} \mathcal {L}(z)\Psi (z)=0, \qquad \Psi :\widetilde{\mathbb {C}^*_J} \rightarrow V(\lambda ). \end{aligned}$$

(3.14)

where $\widetilde{\mathbb {C}^*_J}$ is given by (3.2) and, by abuse of notation, we will denote from now on by z both the global co-ordinate on $\mathbb {C}^*$ the local co-ordinate on $\widetilde{\mathbb {C}^*}$.

Definition 3.3

Given a connection $\mathcal {L}$ of the form (3.1) and a finite dimensional $\widetilde{\mathfrak {g}}$ module V, we say that $\Psi :\widetilde{\mathbb {C}^*_J} \rightarrow V(\lambda )$ is a global solution (or simply, a solution) if $\mathcal {L}(z)\Psi (z)=0$, for all $z \in \widetilde{\mathbb {C}^*_J}$. We denote by $\mathcal {A}_V$ the $\mathbb {C}$-vector space of global solutions of the ODE (3.14).

Definition 3.4

Let $D \subset \widetilde{\mathbb {C}^*_J}$ be open and simply-connected. We say that $\psi : D \rightarrow V(\lambda )$ is a local solution if $\mathcal {L}(z)\psi (z)=0$ for all $z \in D$.

Before addressing the study of global solutions, we remark that a solution $\psi :D \rightarrow V(\lambda )$ is also an analytic function of two variables with domain $D \times \mathbb {C}$: given $\psi$, we denote by $\psi (z;\lambda ) \in V$ the evaluation at the point $\lambda$ of $\psi (z)$. This view-point allows us to make a bridge between a FFH connection and its loop realisation. We have in fact the following

Lemma 3.5

Let $\mathcal {L}$ a connection of the form (3.1), $\mathcal {L}(z,\lambda )$ its loop realisation, $D \subset \widetilde{\mathbb {C}^*_J}$ be open and simply-connected, and V a finite dimensional $\widetilde{\mathfrak {g}}$. The function $\psi :D \rightarrow V(\lambda )$ is a local solution of $\mathcal {L}\psi =0$ if and only if the function ${\widetilde{\psi }}: D \times \mathbb {C} \rightarrow V$, with ${\widetilde{\psi }}(z;\lambda )=\psi (z)(\lambda z^{-k})$, satisfies the differential equation

$$\begin{aligned} \mathcal {L}(z;\lambda ){\widetilde{\psi }}(z,\lambda )=0, \; \psi (z_0;\lambda )= g(\lambda z_0^{-k}), \end{aligned}$$

(3.15)

for all $(z,\lambda ) \in D\times \mathbb {C}$, and it is analytic with respect to the parameter $\lambda$.

Proof

It follows from a direct computation. $\square$

Proposition 3.6

Let V be a finite dimensional $\widetilde{\mathfrak {g}}$-module, $D\subset \widetilde{\mathbb {C}^*_J}$ open and simply-connected, $z_0 \in D$, and $g \in V(\lambda )$. The Cauchy problem

$$\begin{aligned} \Psi (z_0)=g , \qquad \mathcal {L}(z)\Psi (z)=0, \forall z \in D, \end{aligned}$$

(3.16)

where $\mathcal {L}(z)$ is given by (3.1), admits a unique solution $\Psi :D \rightarrow V(\lambda )$.

Proof

Due to Lemma 3.5, the above thesis is equivalent to the statement that for every $\lambda \in \mathbb {C}$ the Cauchy problem

$$\begin{aligned}&\Psi _z(z;\lambda )+\frac{1}{z}\left( \mathring{f}+\ell +(z+z^k\lambda )v_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +X(j) \right) \right) \Psi (z;\lambda )=0. \\&\Psi (z_0;\lambda )=g(\lambda z_0^{-k}) \end{aligned}$$

admits a unique solution ${\widetilde{\psi }}(\cdot ;\lambda ): D \rightarrow V$, and that such a solution depends analytically on $\lambda$.

It is a standard result in the theory of linear ODEs in the complex plane that such a solution exists and it is unique. Moreover, it depends holomorphically on its initial data and on any additional parameters provided the coefficients of the ODE depend holomorphically on them; see e.g. [34, Theorem 1.1]. The thesis follows. $\square$

Definition 3.7

We say that the the connection (3.1) has trivial monodromy or is monodromy-free if for any finite dimensional $\tilde{\mathfrak {g}}$-module V any local solution extends to a global solution.

Definition 3.8

We call Feigin–Frenkel–Hernandez (FFH) connections the class of connections of the form (3.1) which have trivial monodromy.

Definition 3.9

We denote by $\mathcal {O}'$ the sub-ring of the ring of holomorphic function on $\widetilde{\mathbb {C}^*} \times \mathbb {C} \ni (z;\lambda )$ given by

$$\begin{aligned} \mathcal {O}'=\lbrace f(z;\lambda )= Q(z^{-k}\lambda ), \text{ for } \text{ some } \text{ entire } \text{ function } Q: \mathbb {C} \rightarrow \mathbb {C} \rbrace . \end{aligned}$$

(3.17)

Theorem 3.10

The connection $\mathcal {L}$ of the form (3.1) has trivial monodromy if and only if for every finite $\tilde{\mathfrak {g}}$-module V the space $\mathcal {A}_V$ of global solution is a free $\mathcal {O}'$-module of rank $\dim V$ (i.e. $\mathcal {A}_V \cong V \otimes \mathcal {O}'$ as an $\mathcal {O}'$ module).

Proof

The fact that $\mathcal {A}_V$ is an $\mathcal {O}'$-module follows by the fact that $\partial _z -k\textbf{d}/z$ annihilates every function in $\mathcal {O}'$.

Fixed a basis $\lbrace v_1, \dots , v_{\dim V} \rbrace$ of V and a point $z_0 \in \widetilde{\mathbb {C}^*_J}$, we define the local solutions $\psi _i(z)$ via the Cauchy problem $\psi (z_0)=v_i$, $i=1 \dots \dim V$, where we consider $v_i\in V(\lambda )$ as a constant function (in fact, by Proposition 3.6, the local Cauchy problem is well-posed).

Assume that $\mathcal {L}$ has trivial monodromy. By hypothesis the solutions $\lbrace \psi _1, \dots , \psi _{\dim V} \rbrace$ extend to global solutions and, as we prove below, they form an $\mathcal {O}'$ basis of $\mathcal {A}_V$. Let in fact $\psi \in \mathcal {A}_V$. By hypothesis $\psi (z_0) \in V(\lambda )$, namely $\psi (z_0)=\sum _i g_i(\lambda ) v_i$ for some $g_i \in \mathcal {O}_{\lambda }$; hence, by the well-posedness of the local Cauchy problem, $\psi (z)=\sum _{i} g_i(\lambda z^{-k} z_0^{k}) \psi _i(z)$. This prove the only if part of the thesis.

Now assume that V is such that $\mathcal {A}_V$ is a free-module of rank $\dim V$. We show that in this case any local solution extends to a global solution. Let in fact $\psi$ be the solution of the local Cauchy problem $\mathcal {L}\psi =0, \psi (z_0)=g(\lambda )$, for an arbitrary pair $(z_0,g(\lambda ))$. Fixed a basis $\lbrace \varphi _1, \dots , \varphi _{\dim V}\rbrace$ of global solutions, the linear system $\sum _i g_i(\lambda ) \varphi _i(z_0) = g(\lambda )$ admits a unique solution. Therefore locally $\psi (z)=\sum _{i} g_i(\lambda z^{-k} z_0^{k}) \varphi _i(z)$, whence it extends to a global solution. This proves the if part of the thesis. $\square$

Definition 3.11

Let $\mathcal {L} (z;\lambda )$ be the loop realization of $\mathcal {L}(z)$ as per (3.7). We denote by $\mathcal {B}_V$ the $\mathbb {C}$-vector space of analytic functions

$$\begin{aligned} \psi : \widetilde{\mathbb {C}^*_J}\rightarrow V \end{aligned}$$

such that

$$\begin{aligned} \mathcal {L}(z;\lambda )\psi (z;\lambda )=0,\qquad \forall (z,\lambda ) \in \widetilde{\mathbb {C}^*_J} \times \mathbb {C}. \end{aligned}$$

Lemma 3.12

1. The space $\mathcal {B}_V$ is an $\mathcal {O}_{\lambda }-$module and the map

$$\begin{aligned} z^{-k \textbf{d}}: \mathcal {A}_V \rightarrow \mathcal {B}_V , \qquad \psi \mapsto {\widetilde{\psi }}, \end{aligned}$$

(3.18)

where ${\widetilde{\psi }}(z;\lambda )=\psi (z)(\lambda z^{-k})$, is an isomorphism of $\mathbb {C}-$vector spaces.

2. The following properties are equivalent

i):

$\mathcal {L}(z)$ has trivial monodromy.

ii):

For every finite dimensional $\tilde{\mathfrak {g}}$-module V, the space $\mathcal {B}_V$ is a free $\mathcal {O}_{\lambda }-$module of rank $\dim V$ (i.e. $\mathcal {B}_V\cong V(\lambda )$ as $\mathcal {O}_{\lambda }-$modules).

iii):

For every finite dimensional $\tilde{\mathfrak {g}}$-module V, the loop realisation $\mathcal {L}(z;\lambda )$ of $\mathcal {L}(z)$ has trivial monodromy at the singular points $\varepsilon ^l w_j, j \in J$, $l=1,\dots ,r$ for every $\lambda \in {\mathbb {C}}$.

iv):

For every finite dimensional $\tilde{\mathfrak {g}}$-module V, the loop realisation $\mathcal {L}(z;\lambda )$ of $\mathcal {L}(z)$ has trivial monodromy at the singular points $w_j, j \in J$, for every $\lambda \in {\mathbb {C}}$.

Proof

Part 1 follows directly from Lemma 3.5.

Part 1. and Proposition 3.10 imply the equivalence between property i) and property ii). The equivalence between property ii) and property iii) is proved in [45, Lemma 5.4] (for the case $r=1$). The same proof applies verbatim to the case $r>1$. The equivalence between (iii) and (iv) is a consequence of the identity (3.6), namely $\mathcal {L}^{\sigma }_{-\frac{1}{r}}=\mathcal {L}$: The loop realisation of $\mathcal {L}$ has trivial monodromy at $w_j$ if and only if the loop realisation of $\mathcal {L}^{\sigma }$ has trivial monodromy at $w_j$ – as indeed the action of $\mathcal {L}^{\sigma }$ on $V(\lambda )$ coincide with the action of $\mathcal {L}$ on $V^{\sigma }$. Using (3.6), we deduce that the loop realisation of $\mathcal {L}$ has trivial monodromy at $w_j$ if and only if it has trivial at $\varepsilon w_j$. This concludes the proof. $\square$

4 Trivial Monodromy Conditions

In this section, following [45], we study the trivial monodromy conditions the connections (3.1). In Theorem 4.6, we prove that

• If $r=1$, the trivial monodromy conditions are equivalent to a complete system of $|J|(2h-2)$ equations in$|J|(2h-2)$ scalar unknowns;

• If $r>1$, the trivial monodromy conditions are satisfied if and only if $J=\emptyset$.

In order to prove Theorem 4.6, we notice that if $x=z-w_j$ is a local coordinate centered at $w_j$, the connection (3.7) admits an expansion of the form

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{-\theta ^\vee +\eta }{x}+ O (x), \; \eta \in \tilde{\mathfrak {n}}^+ \end{aligned}$$

We show that the property that $\mathcal {L}$ has trivial monodromy at $x=0$ is equivalent to a set of polynomial constraints on the coefficients of the Laurent expansion at $x=0$, which we study to obtain the main theorem of the section. As a preliminary step, we first address the study of the gradation induced by $\theta ^\vee$.

4.1 The gradation induced by $\theta ^\vee$

Let $\theta ^\vee \in \mathring{\mathfrak {h}}\in \mathring{\mathfrak {g}}\subset \tilde{\mathfrak {g}}$ be the element introduced in (2.14) and appearing in the loop realization (3.7). Recall that under the action of $\sigma$ the algebra $\tilde{\mathfrak {g}}$ decomposes as a direct sum of $\mathring{\mathfrak {g}}-$modules: the adjoint $\mathring{\mathfrak {g}}-$module $\tilde{\mathfrak {g}}_0=\mathring{\mathfrak {g}}$, whose weights are the zero weight and the roots of $\mathring{\mathfrak {g}}$, and $r-1$ copies of the $\mathring{\mathfrak {g}}-$module $\tilde{\mathfrak {g}}_1=V_\theta$, whose weights are the zero weight and the short roots of $\mathring{\mathfrak {g}}$. It follows from this that the adjoint action of $\theta ^\vee$ on $\tilde{\mathfrak {g}}$ induces a $\mathbb {Z}-$gradation:

$$\begin{aligned} \tilde{\mathfrak {g}}=\bigoplus _{j\in \mathbb {Z}}\tilde{\mathfrak {g}}_j[\theta ^\vee ],\qquad \tilde{\mathfrak {g}}_j[\theta ^\vee ]=\{x\in \tilde{\mathfrak {g}}\,|\,[\theta ^\vee ,x]=jx\}. \end{aligned}$$

(4.1)

We denote by $\pi _j:\tilde{\mathfrak {g}}\rightarrow \tilde{\mathfrak {g}}_j[\theta ^\vee ]$ the corresponding projection. The gradation (4.1) is compatible with the principal gradation, that is

$$\begin{aligned} \langle \theta ^\vee ,\alpha _i\rangle =-\langle \alpha _0^\vee ,\alpha _i\rangle =-C_{0i}\ge 0,\qquad i\in \mathring{I}, \end{aligned}$$

(4.2)

where we used (2.21). This in particular implies that

$$\begin{aligned} \bigoplus _{j\ge 1}\tilde{\mathfrak {g}}_j[\theta ^\vee ]\subset \tilde{\mathfrak {n}}^+. \end{aligned}$$

(4.3)

Lemma 4.1

(1):

The spectrum of the adjoint action of $\theta ^\vee$ on the $\mathring{\mathfrak {g}}-$module $\tilde{\mathfrak {g}}$ is given in Table 4.

(2):

Let $r>1$, and consider the $\mathring{\mathfrak {g}}-$module $V_\theta =\tilde{\mathfrak {g}}_1\subset \tilde{\mathfrak {g}}$ introduced in Definition 2.1. Then

$$\begin{aligned} V_\theta \cap \tilde{\mathfrak {g}}_2[\theta ^\vee ]=\mathbb {C}v_\theta . \end{aligned}$$

(4.4)

Proof

(1) The weights of the $\mathring{\mathfrak {g}}-$module $\tilde{\mathfrak {g}}$ are the zero weight and the roots of $\mathring{\mathfrak {g}}$. Let ${\widetilde{\theta }}$ be the highest root of $\mathring{\mathfrak {g}}$. A direct case-by-case computation shows that $\langle \theta ^\vee ,{\widetilde{\theta }}\rangle =2$ in all cases except when $\mathfrak {g}=D_4^{(3)}$ (i.e. when $\tilde{\mathfrak {g}}=D_4$ and $\mathring{\mathfrak {g}}=G_2$), in which case $\langle \theta ^\vee ,{\widetilde{\theta }}\rangle =3$. Since the $\theta ^\vee -$gradation is compatible with the principal gradation this value is the maximum of the spectrum. Moreover, the spectrum is symmetric with respect to 0, and all intermediate values are attained except when $\mathfrak {g}=A_1^{(1)}$ (i.e. $\tilde{\mathfrak {g}}=\mathring{\mathfrak {g}}=A_1$) and when $\mathfrak {g}=D_{n+1}^{(2)}$ (i.e. $\tilde{\mathfrak {g}}=D_{n+1}$ and $\mathring{\mathfrak {g}}=B_n$), in which case $\pm 1$ are not part of the spectrum.

(2) The weights of the $\mathring{\mathfrak {g}}-$module $V_\theta$ are the zero weight and the short roots of $\mathring{\mathfrak {g}}$. The highest short root of $\mathring{\mathfrak {g}}$ is $\theta$, and $\langle \theta ^\vee ,\theta \rangle =2$. Reasoning as in part (1) we get that the spectrum is given by $\{-2,0,2\}$ if $\mathfrak {g}=D_{n+1}^{(2)}$, and it is $\{-2,-1,0,1,2\}$ otherwise. To prove (4.4), since the $\theta ^\vee -$gradation is compatible with the principal gradation, it is sufficient to show that $f_iv_\theta \notin \tilde{\mathfrak {g}}_2[\theta ^\vee ]$ for every $i\in \mathring{I}$. To prove that, first notice that

$$\begin{aligned} {[}\theta ^\vee ,f_iv_\theta ]=(2-\langle \theta ^\vee ,\alpha _i\rangle )f_iv_\theta =(2+\langle \alpha _0^\vee ,\alpha _i\rangle )f_iv_\theta =(2+C_{0i})f_iv_\theta . \end{aligned}$$

then $f_iv_\theta \in \tilde{\mathfrak {g}}_2[\theta ^\vee ]$ if and only if $C_{0i}=0$. Assume that $f_iv_\theta \in \tilde{\mathfrak {g}}_2[\theta ^\vee ]$, so that $C_{i0}=C_{0i}=0$. Then, for every $m\in \mathring{I}$ we have

$$\begin{aligned} e_mf_iv_\theta&=f_ie_mv_\theta +[e_m,f_i]v_\theta =\delta _{mi}\alpha ^\vee _i=\delta _{mi}\langle \alpha ^\vee _i,\theta \rangle v_\theta \\&=-\delta _{mi}\langle \alpha ^\vee _i,\alpha _0\rangle v_\theta =-\delta _{mi}C_{i0}v_\theta =0. \end{aligned}$$

Since $V_\theta$ is an irreducible $\mathring{\mathfrak {g}}-$module, we conclude that $f_iv_\theta =0$. $\square$

Example 4.2

Let $\tilde{\mathfrak {g}}=D_3=A_3$, with positive roots ${\tilde{\Delta }}_+=\{{\tilde{\alpha }}_1,{\tilde{\alpha }}_2,{\tilde{\alpha }}_3,{\tilde{\alpha }}_1+{\tilde{\alpha }}_2,{\tilde{\alpha }}_1+{\tilde{\alpha }}_3,{\tilde{\theta }}={\tilde{\alpha }}_1+{\tilde{\alpha }}_2+{\tilde{\alpha }}_3\}$ and basis $\{{\tilde{\alpha }}^\vee _1,{\tilde{\alpha }}^\vee _2,{\tilde{\alpha }}^\vee _3,e_{\pm \alpha }\,| \alpha \in {\tilde{\Delta }}_+\}$, with ${\tilde{\alpha }}^\vee _1$ coroots and $e_{\pm \alpha }$ root vectors. Let $\sigma (1)=1$, $\sigma (2)=3$, $\sigma (3)=2$.

The invariant subalgebra $\mathring{\mathfrak {g}}$ of $\tilde{\mathfrak {g}}$ induced by $\sigma$ is of type $B_2$, with positive roots $\mathring{\Delta }_+=\{\alpha _1={\tilde{\alpha }}_1,\alpha _2=\frac{1}{2}({\tilde{\alpha }}_2+{\tilde{\alpha }}_3),\theta =\alpha _1+\alpha _2,{\tilde{\theta }}=\alpha _1+2\alpha _2\}$ and basis $\{\alpha ^\vee _1,\alpha ^\vee _2,e_{\pm \alpha }\,|\,\alpha \in \mathring{\Delta }_+\}$, where

$$\begin{aligned} \begin{aligned} \alpha ^\vee _{1}&={\tilde{\alpha }}^\vee _1,\\ e_{\pm \alpha _1}&=e_{\pm {\tilde{\alpha }}_1},\\ e_{\pm \theta }&=e_{\pm {\tilde{\alpha }}_1\pm {\tilde{\alpha }}_2}+e_{\pm {\tilde{\alpha }}_1\pm {\tilde{\alpha }}_3} \end{aligned} \qquad \begin{aligned} \alpha ^\vee _2&={\tilde{\alpha }}^\vee _2+{\tilde{\alpha }}^\vee _3,\\ e_{\pm \alpha _2}&=e_{\pm {\tilde{\alpha }}_2}+e_{\pm {\tilde{\alpha }}_3},\\ e_{\pm {\tilde{\theta }}}&=e_{\pm {\tilde{\alpha }}_1\pm {\tilde{\alpha }}_2\pm {\tilde{\alpha }}_3}. \end{aligned} \end{aligned}$$

Since $r=2>1$, the $\mathring{\mathfrak {g}}-$module $V_\theta$ is identified with the subspace $\tilde{\mathfrak {g}}_1\subset \tilde{\mathfrak {g}}$, with basis $\{ v_{-\theta },v_{-\alpha _2},v_0,v_{\alpha _2},v_\theta \}$ where

$$\begin{aligned} v_\theta&=-e_{{\tilde{\alpha }}_1+{\tilde{\alpha }}_2}+e_{{\tilde{\alpha }}_1+{\tilde{\alpha }}_3},\\ v_{\alpha _2}&=[e_{-\alpha _1},v_\theta ]=-e_{{\tilde{\alpha }}_2}+e_{{\tilde{\alpha }}_3},\\ v_0&=[e_{-\alpha _2},v_{\alpha _2}]={\tilde{\alpha }}^\vee _2-{\tilde{\alpha }}^\vee _3,\\ v_{-\alpha _2}&=[e_{-\alpha _2},v_0]=e_{-{\tilde{\alpha }}_2}-e_{-{\tilde{\alpha }}_3},\\ v_{-\theta }&=[e_{-\alpha _1},v_{-\alpha _2}]=e_{-{\tilde{\alpha }}_1-{\tilde{\alpha }}_2}-e_{-{\tilde{\alpha }}_1-{\tilde{\alpha }}_3}. \end{aligned}$$

It is clear from the above identities that $v_\theta$ is a highest weight vector for the $\mathring{\mathfrak {g}}$-module $V_\theta$. Under the action of $\sigma$, the algebra $\tilde{\mathfrak {g}}$ decomposes as $\tilde{\mathfrak {g}}=\tilde{\mathfrak {g}}_0\oplus \tilde{\mathfrak {g}}_1=\mathring{\mathfrak {g}}\oplus V_\theta$, and the element $\theta ^\vee =2\alpha ^\vee _1+\alpha ^\vee _2$ induces on $\tilde{\mathfrak {g}}$ the gradation (4.1), where

$$\begin{aligned} \tilde{\mathfrak {g}}_{-2}[\theta ^\vee ]&=\langle e_{-\alpha _1},e_{-\theta },e_{-{\tilde{\theta }}},v_{-\theta }\rangle ,\\ \tilde{\mathfrak {g}}_0[\theta ^\vee ]&=\tilde{\mathfrak {h}}\oplus \langle v_{-\alpha _2},e_{-\alpha _2},e_{\alpha _2},v_{\alpha _2}\rangle ,\\ \tilde{\mathfrak {g}}_2[\theta ^\vee ]&=\langle e_{\alpha _1},e_\theta ,e_{{\tilde{\theta }}},v_\theta \rangle . \end{aligned}$$

In particular, (4.4) holds true.

We consider the following subspace of $\tilde{\mathfrak {g}}$:

$$\begin{aligned} \mathfrak {u}={{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^++\mathbb {C}v_\theta . \end{aligned}$$

(4.5)

Note that if $r=1$ then $v_\theta =e_\theta \in \mathring{\mathfrak {n}}^+$, and $\mathfrak {u}={{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+$. If $r>1$ the sum in (4.5) is a direct sum. Due to (4.3), and considering the $\sigma$-invariant part only, it follows that

$$\begin{aligned} {{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+=\bigoplus _{j\ge 1}\mathring{\mathfrak {g}}_j[\theta ^\vee ]. \end{aligned}$$

(4.6)

We introduce the following basis of $\mathfrak {u}$. Let $\mathring{\Delta }$ denote the set of roots of $\mathring{\mathfrak {g}}$, and define

$$\begin{aligned} \mathring{\Delta }_{\mathfrak {u}}=\{\alpha \in \mathring{\Delta }|\langle \theta ^\vee ,\alpha \rangle \ge 1\}. \end{aligned}$$

Note the inclusions $\mathring{\Delta }_{\mathfrak {u}}\subset \mathring{\Delta }_+$, where $\mathring{\Delta }_+$ is the set of positive roots of $\mathring{\mathfrak {g}}$ with respect to the Borel subalgebra $\mathring{\mathfrak {b}}^+$. Let $\{e_\alpha \,|\,\alpha \in \mathring{\Delta }_+\}$ be the basis of $\mathring{\mathfrak {n}}^+$ introduced above. Then

$$\begin{aligned} \mathfrak {u}=\left( \bigoplus _{\alpha \in \mathring{\Delta }_{\mathfrak {u}}}\mathbb {C}e_\alpha \right) \oplus \mathbb {C}v_\theta . \end{aligned}$$

(4.7)

4.2 Local conditions on trivial monodromy

If $x=z-w_j$ is a local coordinate centered at $w_j$, the connection (3.7) admits an expansion of the form

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{R}{x}+a+bx+cx^2+O(x^3), \end{aligned}$$

(4.8)

with $R=-\theta ^\vee +\eta$, with $\eta \in \tilde{\mathfrak {n}}^+$ and $a,b,c\in \tilde{\mathfrak {g}}$.

If $\eta _0=\pi _0(\eta )$ is the part of degree 0, with respect to the gradation induced by $\theta ^\vee$, of $\eta$, then $R-\eta _0$ is semi-simple and conjugated to $-\theta ^\vee$, with spectrum provided in Table 4. Therefore $\tilde{\mathfrak {g}}$ splits into eigenspaces of $R-\eta _0$:

$$\begin{aligned} \tilde{\mathfrak {g}}=\bigoplus _i\tilde{\mathfrak {g}}_i[R],\qquad \tilde{\mathfrak {g}}_i[R]=\lbrace x \in \tilde{\mathfrak {g}}, [R-\eta _0,x]= ix \rbrace . \end{aligned}$$

(4.9)

We denote by $\pi ^{R}_{i}:\tilde{\mathfrak {g}}\rightarrow \tilde{\mathfrak {g}}_i[R]$ the corresponding projections.

Definition 4.3

We say that the connection (4.8) has trivial monodromy at 0 if for every finite dimensional $\tilde{\mathfrak {g}}$-module V, the differential equation $\mathcal {L}\psi (x)=0, \psi : D \rightarrow V$, with D a punctured neighborhood of $x=0$, has trivial monodromy.

Proposition 4.4

If $\mathring{\mathfrak {g}}$ is not of type $G_2$, the operator (4.8) has trivial monodromy at $x=0$ if and only if

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

(4.10a)

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

(4.10b)

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

(4.10c)

If $\mathring{\mathfrak {g}}$ is of type $G_2$, the operator (4.8) has trivial monodromy at $x=0$ if and only if

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

(4.11a)

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

(4.11b)

$$\begin{aligned}&\pi ^{R}_{-2}(b)=[\pi ^{R}_{-2}(a),\pi ^{R}_{0}(a)]+\frac{5}{4}[\pi ^{R}_{-3}(a),\pi ^{R}_{1}(a)], \end{aligned}$$

(4.11c)

$$\begin{aligned}&\pi ^{R}_{-3}(c)=\frac{1}{2}[\pi ^{R}_{-3}(a),\pi ^{R}_{0}(b)]+[\pi ^{R}_{-2}(a),\pi ^{R}_{-1}(b)]-[\pi ^{R}_{0}(a),\pi ^{R}_{-3}(b)] \nonumber \\&+\frac{3}{2}[\pi ^{R}_{-2}(a),[\pi ^{R}_{1}(a),\pi ^{R}_{-2}(a)]] +\frac{3}{2} [\pi ^{R}_{0}(a),[\pi ^{R}_{-3}(a),\pi ^{R}_{0}(a)]]. \end{aligned}$$

(4.11d)

Proof

According to the general theory [2, Sec. 3.2], a connection of the form

$$\begin{aligned} \partial _x+\frac{R}{x}+ \sum _{j\ge 0} d_j x^j , \end{aligned}$$

with $d_j\in \tilde{\mathfrak {g}}$, is Gauge equivalent to a connection of the form

$$\begin{aligned} \partial _x+\frac{R}{x}+\sum _{j\ge 0} d'_j x^j , \end{aligned}$$

where $d'_j\in \tilde{\mathfrak {g}}_{-1-j}[R]$. The latter connection is said to be in aligned form and it has trivial monodromy at 0 if and only if $\eta _0=0$ and $d'_j=0$ for all $j \in \text{ spec }(R-\eta _0), j\le -1$.

Explicit computations of the aligned form of the connection lead to the thesis; see [45, Theorem 8.4] for details. $\square$

It follows from the above proposition that if the connection (4.8) has trivial monodromy then $\eta _0=0$. This in turn implies that the coefficient R is semisimple and conjugated to $-\theta ^\vee$; explicitly, we have:

$$\begin{aligned} R=-e^{{{\,\textrm{ad}\,}}{\bar{\eta }}}\theta ^\vee , \end{aligned}$$

(4.12)

where

In the above formula $\eta _i=\pi _i(\eta )$ and $\pi _i$ is the projection associated to the gradation (4.1). The gradation (4.9) is therefore conjugated to the gradation (4.1)

$$\begin{aligned} \tilde{\mathfrak {g}}_i[R]=\lbrace e^{{{\,\textrm{ad}\,}}{\bar{\eta }}}x | x\in \tilde{\mathfrak {g}}_{-i}[\theta ^\vee ], \rbrace \end{aligned}$$

and the relation

$$\begin{aligned} \pi _i^R(x)=e^{{{\,\textrm{ad}\,}}{\bar{\eta }}}(\pi _{-i}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}x)) \end{aligned}$$

holds true for every $x\in \tilde{\mathfrak {g}}$. Using the last identity we can express the trivial monodromy conditions of Proposition 4.4 with respect to the $\theta ^\vee -$gradation:

Proposition 4.5

If $\mathring{\mathfrak {g}}$ is not of type $G_2$, the operator (4.8) has trivial monodromy at $x=0$ if and only if

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

(4.14a)

$$\begin{aligned}&\pi _{1}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)=0, \end{aligned}$$

(4.14b)

$$\begin{aligned}&\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}b)=[\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)], \end{aligned}$$

(4.14c)

where ${\bar{\eta }}$ is given by (4.13a). If $\mathring{\mathfrak {g}}$ is of type $G_2$, the operator (4.8) has trivial monodromy at $x=0$ if and only if

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

(4.15a)

$$\begin{aligned}&\pi _{1}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)=0, \end{aligned}$$

(4.15b)

$$\begin{aligned}&\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}b)=[\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)]+\frac{5}{4}[\pi _{3}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{-1}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)], \end{aligned}$$

(4.15c)

$$\begin{aligned}&\pi _{3}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}c)=\frac{1}{2}[\pi _{3}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}b)]+[\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{1}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}b)]\nonumber \\&-[\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{3}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}b)]+\frac{3}{2}[\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),[\pi _{-1}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)]] \nonumber \\&+\frac{3}{2} [\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),[\pi _{3}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a),\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}}a)]], \end{aligned}$$

(4.15d)

where ${\bar{\eta }}$ is given by (4.13b).

4.3 Monodromy equations for FFH connections

After Lemma 3.12, the FFH connection (3.1) has trivial monodromy if and only if its loop realisation $\mathcal {L}(z,\lambda )$ has trivial monodromy at all $w_j$, $j \in J$, for every $\lambda$. According to Proposition 4.5, $\mathcal {L}(z,\lambda )$ has trivial monodromy at $w_j$ if and only if its Laurent expansion at $w_j$ satisfies the constraints (4.14)—or (4.15) in the case $\mathring{\mathfrak {g}}=\mathfrak {g}_2$—for every $\lambda \in \mathbb {C}$. Therefore, the FFH connection (3.1) has trivial monodromy if and only if the constraints (4.14)—or (4.15)—are satisfied at all $w_j$, $j \in J$, and for every $\lambda \in \mathbb {C}$. To proceed further, we write these constraints explicitly in terms of the connection $\mathcal {L}(z;\lambda )$, as in (3.7). Fix $j\in J$ and let $x=z-w_j$, the connection $\mathcal {L}(z;\lambda )$ admits an expansion of the form (4.8), say

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{R(j)}{x}+a(j)+b(j)x+c(j)x^2+O(x^3), \end{aligned}$$

(4.16)

with $R(j)=-\theta ^\vee +\eta (j)$, with $\eta (j) \in \tilde{\mathfrak {n}}^+$ and $a(j),b(j),c(j)\in \tilde{\mathfrak {g}}$. The explicit form of $\eta (j)$, a(j) and b(j), for $j\in J$, is given by:

$$\begin{aligned} \eta (j)=&X(j), \end{aligned}$$

(4.17a)

$$\begin{aligned} w_ja(j)=&\mathring{f}+\ell +(w_j+w_j^k\lambda )v_\theta +\frac{r-1}{2}(-\theta ^\vee +X(j))\nonumber \\&+\sum _{i\ne j}\frac{rw_j^r}{w_j^r-w_i^r}\left( -\theta ^\vee +X(i)\right) , \end{aligned}$$

(4.17b)

$$\begin{aligned} w_j^2b(j)=&-\mathring{f}-\ell +(k-1)w_j^k\lambda v_\theta +\frac{(r-1)(r-5)}{12}(-\theta ^\vee +X(j))\nonumber \\&-\sum _{i\ne j}\frac{rw_j^r\left( (r-1)w_i^r+w_j^r)\right) }{w_j^r-w_i^r}(-\theta ^\vee +X(i)) \end{aligned}$$

(4.17c)

The term c(j), which is relevant only in the $\mathfrak {g}=D_4^{(3)}$ case, can be computed in a similar way. The coefficients a(j), b(j) and c(j) depend on the loop variable $\lambda$, while $\eta (j)$ does not. For each $j\in J$ we consider the trivial monodromy system (4.14) (if $\mathring{\mathfrak {g}}$ is not of type $G_2$) or (4.15) (if $\mathring{\mathfrak {g}}$ is of type $G_2$). These equations have to be satisfied for every value of $\lambda$. For example, if $\mathfrak {g}\ne D_4^{(3)}$, then equations (4.14) applied to the present case read:

$$\begin{aligned}&\pi _0(\eta (j))=0, \end{aligned}$$

(4.18a)

$$\begin{aligned}&\pi _{1}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}(j)}a(j))=0, \end{aligned}$$

(4.18b)

$$\begin{aligned}&\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}(j)}b(j))=[\pi _{2}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}(j)}a(j)),\pi _{0}(e^{-{{\,\textrm{ad}\,}}{\bar{\eta }}(j)}a(j))], \end{aligned}$$

(4.18c)

where ${\bar{\eta }}(j)$ is given by (4.13a):

$$\begin{aligned} {\bar{\eta }}(j)=\pi _1(\eta (j))+\frac{1}{2}\pi _2(\eta (j)). \end{aligned}$$

The unknowns of the trivial monodromy system (4.18) are $w_j\in \mathbb {C}^*$, and $X(j)\in \tilde{\mathfrak {n}}^+\cap \mathring{\mathfrak {g}}$, or equivalently by $\{w_j,\eta (j)|j\in J\}$, where $\eta (j)$ is given in (4.17a). Recalling the definition (4.5) of the subspace $\mathfrak {u}$, and using (4.18a) together with (4.6) we obtain

$$\begin{aligned} \eta (j)\in \mathfrak {u}\cap \mathring{\mathfrak {n}}^+={{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+, \end{aligned}$$

and we can write $\eta (j)$ with respect to the basis (4.7) of $\mathfrak {u}$ as

$$\begin{aligned} \eta (j)=\sum _{\alpha \in \mathring{\Delta }_{\mathfrak {u}}}x^\alpha (j)e_\alpha , \end{aligned}$$

for some $x^\alpha (j)\in \mathbb {C}$. Given the connection (3.1), the trivial monodromy system (4.18) turns into a system of algebraic equations for the complex unknowns

$$\begin{aligned} U_J= \lbrace w_j,x^{\alpha }(j)\,|\,\alpha \in \mathring{\Delta }_{\mathfrak {u}}\rbrace _{j \in J}, \end{aligned}$$

(4.19)

which we analyse below. Note, incidentally, that $|U_J|=|J|(\dim ({{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+)+1)$, where |J| is the cardinality of J. It then follows that

$$\begin{aligned} |U_J|= {\left\{ \begin{array}{ll} |J|(\dim \mathfrak {u}+1),\qquad &{}r=1,\\ |J|\dim \mathfrak {u},\qquad &{}r>1. \end{array}\right. } \end{aligned}$$

(4.20)

Theorem 4.6

Let $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$ be as in Table 4.

• (1) If $r=1$, the connection (3.1) has trivial monodromy if and only if the $|J|(\dim \mathfrak {u}+1)$ scalar unknowns $U_J=\{w_j,x^\alpha (j)|\alpha \in \mathring{\Delta }_{\mathfrak {u}},j\in J\}$ satisfy the system of $|J|(\dim \mathfrak {u}+1)$ equations (4.18) with $j\in J$. In this case, $\dim \mathfrak {u}=2h-3$, where h is the Coxeter number of $\mathfrak {g}$.

• (2) If $r>1$, the connection (3.1) has trivial monodromy if and only if $J=\emptyset$.

Proof

(1) Proved in [45]. Note that in this case $v_\theta =e_\theta \in \mathring{\mathfrak {g}}=\tilde{\mathfrak {g}}$ and $\theta$ is the highest root of $\mathring{\mathfrak {g}}$, so that $\mathfrak {u}={{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+$.

(2) We consider the loop realization (3.7) of the oper (3.1), and assume that $\mathfrak {g}\ne D_4^{(3)}$, so that the trivial monodromy equations are given by (4.18). Since $r>1$ we have $\mathfrak {u}={{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+\oplus \mathbb {C}v_\theta$, and the set of unknowns $U_J$ of the trivial monodromy sistem (4.18) consists of $|J|\dim \mathfrak {u}$ scalar variables $\lbrace w_j,x^{\alpha }(j)\,|\,\alpha \in \mathring{\Delta }_{\mathfrak {u}}\rbrace _{j \in J}$. To count the number of equations we write (4.18) in a more explicit form. We fix $j\in J$ and for simplicity we drop the dependence on j in the following formulae and we denote $x_i=\pi _i(x)$ ($x=\eta ,a,b$). For $\mathfrak {g}$ of type $D_{n+1}^{(2)}$, the spectrum of $\theta ^\vee$ is equal to $\{-2,0,2\}$ and system (4.18) is equivalent to

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

(4.21a)

$$\begin{aligned}&b_2-\frac{1}{2}[\eta _2,b_{0}]+\frac{1}{8}[\eta _2,[\eta _2,b_{-2}]] \nonumber \\ {}&\quad = [a_2-\frac{1}{2}[\eta _2,a_{0}]+\frac{1}{8}[\eta _2,[\eta _2,a_{-2}]], a_0-\frac{1}{2}[\eta _2,a_{-2}]], \end{aligned}$$

(4.21b)

Condition (4.21a) is satisfied by choosing $\eta (j)=X(j)\in {{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+\subsetneq \mathfrak {u}$. Equation (4.21b) consists of a $\sigma$-invariant part, which is constant in $\lambda$ and provides $\dim ({{\,\textrm{ad}\,}}_{\theta ^\vee }\mathring{\mathfrak {n}}^+)$ equations, and a $\sigma$ skew invariant part, which depends linearly on $\lambda$ and it is given by

$$\begin{aligned} (k-1)w_j^{k-2}\lambda v_\theta =\left[ (w_j+w_j^{k-1}\lambda ) v_\theta ,a_0(j)-\frac{1}{2}[\eta _2(j),a_{-2}(j)]\right] . \end{aligned}$$

(4.22)

Since $a_0(j),\eta _2(j)$ and $a_{-2}(j)$ all belong to $\mathring{\mathfrak {g}}\cap \tilde{\mathfrak {g}}_0[\theta ^\vee ]$, the right hand side in (4.22) belongs to $V_\theta \cap \tilde{\mathfrak {g}}_2[\theta ^\vee ]=\mathbb {C}v_\theta$, where we used (4.4). It follows that (4.22) can be written as

$$\begin{aligned} (k-1)w_j^{k-2}\lambda =(w_j+w_j^{k-1}\lambda )C(j), \end{aligned}$$

(4.23)

for some scalar function C(j), and a simple computation shows that

$$\begin{aligned} C(j)=-\ell (\theta )+1+\sum _{i\ne j}\frac{4w_j^2}{w_j^2-w_i^2}. \end{aligned}$$

Condition (4.23) has to be satisfied for every value of $\lambda$. The constant part in $\lambda$ gives the condition $w_j C(j)=0$ which implies $C(j)=0$ since $w_j\in \mathbb {C}^*$. Substituting back into (4.23) we obtain $k=1$ which is not acceptable since $k\in (0,1)$.

If $\mathfrak {g}$ is of type $A_{2n-1}^{(2)}$ or $E_6^{(2)}$, the spectrum of $\theta ^\vee$ is given by $\{-2,-1,0,1,2\}$ and – by direct inspection at (4.17)—we have $a_{-2}=b_{-2}=0$. It follows that (4.18) is equivalent to

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

(4.24a)

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

(4.24b)

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

(4.24c)

The $\sigma$ skew-invariant part is contained in (4.24c) only, and explicitely reads

$$\begin{aligned} (k-1)w_j^{k-2}\lambda v_\theta =\left[ (w_j+w_j^{k-1}\lambda ) v_\theta ,a_0(j)-\frac{1}{2}[\eta _1(j),a_{-1}(j)]\right] . \end{aligned}$$

(4.25)

Reasoning as in the previous case, we obtain again condition (4.23), which leads to a contradiction.

The proof for $D_4^{(3)}$ is similar, the only difference being the trivial monodromy equations are given by (4.15). $\square$

4.4 Monodromy equation for extended FFH $\mathfrak {g}$-opers

Due to the negative result proved in Theorem 4.6, it is natural to ask whether there exists a more general class of opers, which still provide solutions to the QQ-system (5.38) and for which the trivial monodromy conditions are non-empty. We thus consider connections of the form

$$\begin{aligned} \mathcal {L}(z)=&\partial _z+\frac{1}{z}\left( f+\ell -k \textbf{d}+ zv_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +\sum _{m=0}^{r-1}\left( \frac{z}{w_j}\right) ^mX_m(j), \right) \right) . \end{aligned}$$

(4.26)

where

$$\begin{aligned} X_m(j)\in \tilde{\mathfrak {g}}_m,\qquad m=0,\dots ,r-1. \end{aligned}$$

Their loop realization is given by

$$\begin{aligned} \mathcal {L}(z;\lambda )=&\partial _z+\frac{1}{z}\left( f+\ell + (z+z^k\lambda )v_\theta + \sum _{j \in J} \frac{ r z^{r}}{z^r-w_j^r}\left( -\theta ^\vee +\sum _{m=0}^{r-1}\left( \frac{z}{w_j}\right) ^mX_m(j), \right) \right) . \end{aligned}$$

(4.27)

These connections are not of the form suggested in [30]; however they coincide with them when $J=\emptyset$ (they describe the same ground-state oper), and they a provide solutions to the same QQ system (5.38), if the monodromy is trivial about all $w_j, j \in J$. In fact, since the additional terms are subdominant at 0 and $\infty$ and satisfies the symmetry (3.6), the theory that we develop in Sect. 5 would still apply to these more general connections. Moreover, as we will study in Sect. 6, they are still representatives of twisted parabolic Miura $\mathfrak {g}$-opers (see (A.37)). We now prove that even in this more general class of connections, the trivial monodromy equations have no solutions.

The construction is similar to the one described above: we consider the following subspace of $\tilde{\mathfrak {g}}$:

$$\begin{aligned} \tilde{\mathfrak {u}}={{\,\textrm{ad}\,}}_{\theta ^\vee }\tilde{\mathfrak {n}}^+, \end{aligned}$$

(4.28)

and due to (4.3) it follows that

$$\begin{aligned} \tilde{\mathfrak {u}}=\bigoplus _{j\ge 1}\tilde{\mathfrak {g}}_j[\theta ^\vee ]. \end{aligned}$$

(4.29)

Since $\mathring{\mathfrak {g}}\subset \tilde{\mathfrak {g}}$, we have

$$\begin{aligned} \mathfrak {u}\subset \tilde{\mathfrak {u}}, \end{aligned}$$

and the inclusion is strict if $r>1$. We introduce the following basis of $\tilde{\mathfrak {u}}$. Let $\mathring{\Delta }=\mathring{\Delta }_{\text {long}}\cup \mathring{\Delta }_{\text {short}}$ be the decomposition of the set of roots of $\mathring{\mathfrak {g}}$ into long and short roots. Define

$$\begin{aligned} \mathring{\Delta }_{\tilde{\mathfrak {u}}}=\{\alpha \in \mathring{\Delta }|\langle \theta ^\vee ,\alpha \rangle \ge 1\}, \end{aligned}$$

and

$$\begin{aligned} \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}=\mathring{\Delta }_{\tilde{\mathfrak {u}}}\cap \mathring{\Delta }_{\text {short}} \end{aligned}$$

Note the inclusions $\mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}\subset \mathring{\Delta }_{\tilde{\mathfrak {u}}}\subset \mathring{\Delta }_+$, where $\mathring{\Delta }_+$ is the set of positive roots of $\mathring{\mathfrak {g}}$ with respect to the Borel subalgebra $\mathring{\mathfrak {b}}^+$. Let $\{e_\alpha \,|\,\alpha \in \mathring{\Delta }_+\}$ be the basis of $\mathring{\mathfrak {n}}^+$ introduced above. Note in particular that $\theta \in \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}$, so the element $v_\theta$ appearing in (4.27) belongs to $\tilde{\mathfrak {u}}$. We can, and will from now on, fix elements $\{v_\alpha \,|\,\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}\}\subset \tilde{\mathfrak {g}}_1$ (if $r>1$) and $\{{\bar{v}}_\alpha \,|\,\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}\}\subset \tilde{\mathfrak {g}}_2$ (if $r>2$) such that

$$\begin{aligned} \tilde{\mathfrak {u}}= {\left\{ \begin{array}{ll} {\bigoplus }_{\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}}}}\mathbb {C}e_\alpha ,\quad &{} r=1,\\ \left( {\bigoplus }_{\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}}}}\mathbb {C}e_\alpha \right) \oplus \left( {\bigoplus }_{\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}}\mathbb {C}v_\alpha \right) ,\quad &{} r=2,\\ \left( {\bigoplus }_{\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}}}}\mathbb {C}e_\alpha \right) \oplus \left( {\bigoplus }_{\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}}\mathbb {C}v_\alpha \right) \oplus \left( {\bigoplus }_{\alpha \in \mathring{\Delta }_{\tilde{\mathfrak {u}},\text {short}}}\mathbb {C}{\bar{v}}_\alpha \right) ,\quad &{} r=3. \end{array}\right. } \end{aligned}$$

(4.30)

Fix now $j\in J$ and let $x=z-w_j$, the connection $\mathcal {L}(z;\lambda )$ admits an expansion of the form (4.8), say

$$\begin{aligned} \mathcal {L}=\partial _x+\frac{R(j)}{x}+a(j)+b(j)x+c(j)x^2+O(x^3), \end{aligned}$$

(4.31)

with $R(j)=-\theta ^\vee +\eta (j)$, with $\eta (j) \in \tilde{\mathfrak {n}}^+$ and $a(j),b(j),c(j)\in \tilde{\mathfrak {g}}$. The explicit form of $\eta (j)$, a(j) and b(j), for $j\in J$, is given by:

$$\begin{aligned} \eta (j)=&\sum _{m=0}^{r-1}X_m(j), \end{aligned}$$

(4.32a)

$$\begin{aligned} w_ja(j)=&\mathring{f}+\ell +(w_j+w_j^k\lambda )v_\theta -\frac{r-1}{2}\theta ^\vee +\sum _{m=0}^{r-1}\frac{r-1+2m}{2}X_m(j)\nonumber \\&+\sum _{i\ne j}\frac{rw_j^r}{w_j^r-w_i^r}\left( -\theta ^\vee +\sum _{m=0}^{r-1}\left( \frac{w_j}{w_i}\right) ^mX_m(i)\right) , \end{aligned}$$

(4.32b)

$$\begin{aligned} w_j^2b(j)=&-\mathring{f}-\ell +(k-1)w_j^k\lambda v_\theta -\frac{(r-1)(r-5)}{12}\theta ^\vee \nonumber \\&+\sum _{m=0}^{r-1}\frac{6m(m+r-2)+(r-1)(r-5)}{12}X_m(j) \end{aligned}$$

(4.32c)

$$\begin{aligned}&+\sum _{i\ne j}\frac{rw_j^r\left( (r-1)w_i^r+w_j^r)\right) }{w_j^r-w_i^r}\theta ^\vee \nonumber \\&-\sum _{i\ne j}\frac{rw_j^r}{w_j^r-w_i^r}\left( \sum _{m=0}^{r-1}\frac{(rw_i^r+(m-1)(w_i^r-w_j^r))w_j^m}{w_i^m}X_m(i)\right) \end{aligned}$$

(4.32d)

The term c(j), which is relevant only in the $\mathfrak {g}=D_4^{(3)}$ case, can be computed in a similar way. The coefficients a(j), b(j) and c(j) depend on the loop variable $\lambda$, while $\eta (j)$ does not. For each $j\in J$ we consider the trivial monodromy system (4.14) (if $\mathring{\mathfrak {g}}$ is not of type $G_2$) or (4.15) (if $\mathring{\mathfrak {g}}$ is of type $G_2$). These equations have to be satisfied for every value of $\lambda$.

The unknowns of the trivial monodromy system (4.18) are $w_j\in \mathbb {C}^*$, and $X_m(j)\in \tilde{\mathfrak {n}}^+\cap \tilde{\mathfrak {g}}_m$, with $m=0,\dots ,r-1$, or equivalently by $\{w_j,\eta (j)|j\in J\}$, where $\eta (j)$ is given in (4.32a). Recalling the definition (4.28) of the subspace $\tilde{\mathfrak {u}}$, and using (4.18a) together with (4.29) we obtain

$$\begin{aligned} \eta (j)\in \tilde{\mathfrak {u}}, \end{aligned}$$

and we can write $\eta (j)$ with respect to the basis (4.30) of $\tilde{\mathfrak {u}}$ as

$$\begin{aligned} \eta (j)=\sum _{\alpha \in \mathring{\Delta }_{\mathfrak {u}}}x^\alpha (j)e_\alpha +(r-1) \sum _{\alpha \in \mathring{\Delta }_{\mathfrak {u},\text {short}}}y^\alpha (j)v_\alpha +(r-1)(r-2) \sum _{\alpha \in \mathring{\Delta }_{\mathfrak {u},\text {short}}}{\bar{y}}^\alpha (j){\bar{v}}_\alpha , \end{aligned}$$

for some $x^\alpha (j),y^\alpha (j),{\bar{y}}^\alpha (j)\in \mathbb {C}$. Hence, (4.18) turns into a system of algebraic equations for the complex unknowns

$$\begin{aligned} {\widetilde{U}}_J= {\left\{ \begin{array}{ll} \lbrace w_j,x^{\alpha }(j)\,|\,\alpha \in \mathring{\Delta }_{\mathfrak {u}}\rbrace _{j \in J},\qquad &{} r=1,\\ \lbrace w_j,x^{\alpha }(j),y^\beta (j)\,|\,\alpha \in \mathring{\Delta }_{\mathfrak {u}},\beta \in \mathring{\Delta }_{\mathfrak {u},\text {short}}\rbrace _{j \in J},\qquad &{} r=2,\\ \lbrace w_j,x^{\alpha }(j),y^\beta (j),{\bar{y}}^\gamma (j)\,|\,\alpha \in \mathring{\Delta }_{\mathfrak {u}},\beta ,\gamma \in \mathring{\Delta }_{\mathfrak {u},\text {short}}\rbrace _{j \in J},\qquad &{}r=3. \end{array}\right. } \end{aligned}$$

(4.33)

Note that the number of scalar unknowns is

$$\begin{aligned} |{\widetilde{U}}_J|=|J|(\dim \tilde{\mathfrak {u}}+1). \end{aligned}$$

In order to obtain a complete system of equations, we expect to obtain the same number of equations in the same number of unknowns. If $r>1$ the first negative answer is provided by the following:

Proposition 4.7

If $\mathfrak {g}$ is of type $D_{n+1}^{(2)}$, the connection (4.26) has trivial monodromy if and only if $J=\emptyset$.

Proof

Let $\mathfrak {g}$ be of type $D_{n+1}^{(2)}$, so that the trivial monodromy system (4.14) is given by (4.21). Condition (4.21a) is satisfied by choosing $\eta (j)\in \mathfrak {u}$, or equivalently, due to (4.17a), choosing $X_m(j)\in \tilde{\mathfrak {u}}\cap \tilde{\mathfrak {g}}_m$, $m=0,\dots ,r-1$. Equation (4.21b) is linear in $\lambda$: the constant part provides $|J|\dim \mathfrak {u}$ equations, while the linear part reads:

$$\begin{aligned} (k+1)w_j^{k-2}v_\theta =\left[ w_j^{k-1}v_\theta ,a_0(j)-\frac{1}{2}[\eta _2(j),a_{-2}(j)]\right] . \end{aligned}$$

(4.34)

Since $X_m(j)\in \tilde{\mathfrak {u}}$ we get from (4.32b) that $a_0(j)\in \mathring{\mathfrak {g}}$, and therefore $[v_\theta ,a_0(j)]\in V_\theta \cap \tilde{\mathfrak {g}}_2[\theta ^\vee ]=\mathbb {C}v_\theta$. The projection onto $\mathbb {C}v_\theta$ of (4.34) thus provides additional |J| equations, which brings the total number of equations to $|J|(\dim \tilde{\mathfrak {u}}+1)$. However, the element $\eta _2(j)$ in (4.34) contains the term $w_j^{-1}y^\theta (j)v_\theta$, and since in this case $a_{-1}(j)=f_1$, the term $[w_j^{k-1}v_\theta ,[w_j^{-1}y^\theta (j)v_\theta ,f_1]]$ appears. For $\mathring{\mathfrak {g}}$ of type $B_n$, $\theta -\alpha _1$ is a short root and $\theta +(\theta -\alpha _1)={\tilde{\theta }}$, the highest root of $B_n$. We therefore obtain the additional condition

$$\begin{aligned} w_j^{k-2}y^\theta (j)e_{{\tilde{\theta }}}=0. \end{aligned}$$

(4.35)

From this equation we obtain $y^\theta (j)=0$, and substituting into (4.21b) we get $w_j=0$, which is not acceptable since $w_j\in \mathbb {C}^*$. $\square$

Remark 4.8

We make some comments on the other cases. For $\mathfrak {g}$ of type $A_{2n-1}^{(2)}$, we have $r=2$ and $\mathring{\mathfrak {g}}=C_n$. The trivial monodromy system (4.18) is given by (4.24). Condition (4.24a) is equivalent to $\eta \in \tilde{\mathfrak {u}}$, so it is identically satisfied. The dependence on $\lambda$ of the system (4.24) is only through the terms $a_2$ and $b_2$, which are linear in $\lambda$. Equation (4.24b) and the constant (in $\lambda$) term of (4.24c) take values in $\tilde{\mathfrak {u}}$, leading to $|J|\dim \tilde{\mathfrak {u}}$ equations. The linear term in $\lambda$ of equation (4.24c) reads

$$\begin{aligned} (k-1)w_j^{k-2}v_\theta =\left[ w_j^{k-1}v_\theta ,a_0(j)-[\eta _1(j),a_{-1}(j)]\right] . \end{aligned}$$

(4.36)

As before, $a_0(j)\in \mathring{\mathfrak {g}}$ and therefore $[v_\theta ,a_0(j)]\in V_\theta \cap \tilde{\mathfrak {g}}_2[\theta ^\vee ]=\mathbb {C}v_\theta$. Projecting (4.34) (for $j\in J$) onto $\mathbb {C}v_\theta$ gives additional |J| equations, which brings the total number of equations to $|J|(\dim \tilde{\mathfrak {u}}+1)$. However, for $\mathring{\mathfrak {g}}$ of type $C_n$ we have that $\alpha _1+\alpha _2$ is a short root satisfying $\langle \theta ^\vee ,\alpha _1+\alpha _2\rangle =1$ so that the element $\eta _1(j)$ contains the term $y^{\alpha _1+\alpha _2}(j)v_{\alpha _1+\alpha _2}$. In this case $a_{-1}(j)=f_2$ so that in (4.36) the term $\left[ w_j^{k-1}v_\theta ,[y^{\alpha _1+\alpha _2}(j)v_{\alpha _1+\alpha _2},f_2]\right]$ appears. In addition $\theta +(\alpha _1+\alpha _2-\alpha _2)={\tilde{\theta }}$, the highest root of $C_n$, and we obtain the additional condition

$$\begin{aligned} w_j^{k-1}y^{\alpha _1+\alpha _2}(j)e_{{\tilde{\theta }}}=0. \end{aligned}$$

(4.37)

We considered system (4.21) in the in the particular case when $\mathfrak {g}$ of type $A_5^{(2)}$, obtaining again no solutions. The computations are more involved and will be omitted. The idea is that from the additional condition (4.37) one obtains $y^{\alpha _1+\alpha _2}(j)=0$: this time the substitution into (4.21b) does not give an immediate contradiction, but considering the full system (4.21) eventually one obtains that there are no solutions.

For $\mathfrak {g}$ of type $E_6^{(2)}$ the trivial monodromy system is again (4.24). The linear term in $\lambda$ is given by (4.36), and also in this case we get an additional condition, as follows: for $\mathring{\mathfrak {g}}$ of type $F_4$ the root $\beta =\alpha _1+\alpha _2+\alpha _3+\alpha _4$ is short and satisfies $\langle \theta ^\vee ,\beta \rangle =1$. Therefore, $\eta _1(j)$ contains the term $y^{\beta }(j)v_{\beta }$. In this case $a_{-1}(j)=f_1$ so that in (4.36) the term $\left[ w_j^{k-1}v_\theta ,[y^{\beta }(j)v_{\beta },f_1]\right]$ appears. In addition, $\beta -\alpha _2$ is a (short) root, and $\theta +(\beta -\alpha _2)={\tilde{\theta }}$, the highest root of $F_4$. We thus obtain the additional condition $w_j^{k-1}y^{\beta }(j)e_{{\tilde{\theta }}}=0$.

The proof for $D_4^{(3)}$ is similar. Due to the above discussion, we expect the same negative result of Proposition 4.7 to hold for all other $r>1$ cases.

5 Derivation of the QQ System

In this section, we show that the monodromy data (or generalised spectral determinants) of FFH connections provide solutions to the Bethe equations for the quantum $^L\mathfrak {g}$-Drinfeld–Sokolov model. The result of this section builds on our previous paper [45,46,47]. Here however, we give a complete and unified theory, and we fill important gaps in the literature.

We recall here that, due to Theorem 4.6, if the FFH connection 3.1 has trivial monodromy and $r>1$, then $J =\emptyset$. For this reason, without loss in generality, the FFH connection $\mathcal {L}$ 3.1 and the domain of analiticity 3.2 of the general solution of the differential equation $\mathcal {L}\psi =0$ admit the simplified forms

$$\begin{aligned}&\mathcal {L}(z)= \partial _z+\frac{1}{z}\left( f+\ell -k \textbf{d}+ zv_\theta + \sum _{j \in J} \frac{ z}{z-w_j}\left( -\theta ^\vee +X(j) \right) \right) , \end{aligned}$$

(5.1)

$$\begin{aligned}&\widetilde{\mathbb {C}^*_J}=\widetilde{\mathbb {C}^*} \setminus \bigcup _{j \in J} \Pi ^{-1} ( w_j), \end{aligned}$$

(5.2)

which we will use throughout this section.

5.1 Rotated and twisted solutions

Let $\mathcal {L}$ be a Feigin–Frenkel–Hernandez connection, and for $t\in \mathbb {R}$ consider the rotated connection $\mathcal {L}_t$, given by (3.4). Fixed $V(\lambda )$ as in (3.13) consider the ODE

$$\begin{aligned} \mathcal {L}_t(z)\Psi (z)=0, \quad \Psi :\left( \widetilde{\mathbb {C}^*} \setminus \bigcup _{j \in J}\Pi ^{-1} (e^{-2\pi i t} w_j)\right) \rightarrow V(\lambda ),\end{aligned}$$

and denote by $\mathcal {A}_{V,t}$ be the corresponding space of solutions. Denoting

$$\begin{aligned} \psi _t(z)= \psi (e^{2\pi i t}z), \end{aligned}$$

then the map

$$\begin{aligned} \mathcal {A}_V \rightarrow \mathcal {A}_{V,t}, \qquad \psi \mapsto \psi _t, \end{aligned}$$

is an isomorphism of $\mathbb {C}$-vector spaces. Moreover, since $\mathcal {L}$ is single-valued, for every $t\in \mathbb {R}$ the spaces $\mathcal {A}_{V,t}$ and $\mathcal {A}_{V,t+1}$ actually coincide. Therefore there is a $\mathbb {C}$-automorphism $\mathcal {M}$ of $\mathcal {A}_{V,t}$, called the monodromy (or monodromy operator) and defined by

$$\begin{aligned} \mathcal {M}: \mathcal {A}_{V,t} \rightarrow \mathcal {A}_{V,t} ,\qquad \psi \mapsto \psi _{1} . \end{aligned}$$

(5.3)

If $L({\tilde{\omega }}_i)$ ($i \in {\widetilde{I}}$) is the i-th fundamental $\widetilde{\mathfrak {g}}$-module, we denote

$$\begin{aligned} \mathcal {A}^{(i)}:=\mathcal {A}_{L({\tilde{\omega }}_i),\kappa _i}, \quad \mathcal {A}^{(i)}_t=\mathcal {A}_{L({\tilde{\omega }}_i),\kappa _i+t} \end{aligned}$$

(5.4)

where $\kappa _i$’s are the rational numbers defined in Table 3.

Now recall the algebra $\widetilde{\mathfrak {g}}$ carries a Dynkin automorphism $\sigma$ and that for any representation V we have defined the linear isomorphism (2.29) from V to the twisted module $V^{\sigma }$. Since FFH connections satisfy (3.6), we have an isomorphism between the spaces of solutions $\mathcal {A}_V$ and $\mathcal {A}_{V^\sigma ,-\frac{1}{r}}$. More precisely, we have

Lemma 5.1

1. For every finite dimensional module V and every $t\in \mathbb {R}$, the map (2.29) induces a $\mathcal {O}'-$isomorphism between $\mathcal {A}_{V,t}$ and $\mathcal {A}_{V^{\sigma },t+\frac{1}{r}}$. In particular, for every $i \in {\tilde{I}}$,

$$\begin{aligned}&\Psi \in \mathcal {A}^{(i)}_t \Longleftrightarrow \sigma (\Psi ) \in \mathcal {A}^{(\sigma (i))}_{t+\frac{1}{r}} \end{aligned}$$

(5.5)

$$\begin{aligned}&\Psi \in \mathcal {A}^{(i)}_t \Longleftrightarrow R_i(\Psi ) \in \mathcal {A}^{(i)}_{t+ D_i} \end{aligned}$$

(5.6)

In the latter equation $R_i: L({\tilde{\omega }}_i) \rightarrow L({\tilde{\omega }}_i)$ is given by (2.33) and the coefficients $D_i$ by (2.8).

2. For $i\in \mathring{I}$ and $t\in \mathbb {R}$ the map ${\widetilde{m}}_i$ defined in (2.27) induces a $\mathbb {C}-$linear map

$$\begin{aligned} {\widetilde{m}}_i:\bigwedge ^2 \mathcal {A}^{(i)}_t\rightarrow \bigotimes _{j\in \mathring{I}}\bigotimes _{\ell =0}^{\langle j\rangle -1}\left( \mathcal {A}^{\sigma ^\ell (j)}_{\kappa _i-\kappa _j+t}\right) ^{\otimes {\bar{B}}_{i\sigma ^\ell (j)}} \end{aligned}$$

Using (5.5) there exists a $\mathbb {C}-$linear map

$$\begin{aligned} m_i:\bigwedge ^2 \mathcal {A}^{(i)}_{\frac{D_i}{2}}\longrightarrow \bigotimes _{j\in \mathring{I}} \bigotimes _{l=0}^{B_{ij}-1}\mathcal {A}^{(j)}_{\frac{B_{ij}-1-2l}{2r}}\,, \qquad i\in \mathring{I}, \end{aligned}$$

(5.7)

which has the property that

$$\begin{aligned} m_i(v_i \wedge f_iv_i) = \otimes _{j \in \mathring{I}} v^{\otimes B_{ij}}_j, \end{aligned}$$

(5.8)

where $v_j$ is a highest weight vector of $L({\tilde{\omega }}_j)$.

Proof

1. is a straightforward check and 2. is proved in [47, Proposition 4.5]. $\square$

5.1.1 Frobenius solutions: a basis of monodromy eigenvectors

We discuss Frobenius-like solutions of the ODE (3.14). From these solutions, under some genericity conditions on the parameters $\ell , k$ we will construct a basis of eigenvectors for the monodromy operator (5.3). We closely follow [45, Section 5.2], to which we refer for further details and proofs.

A Frobenius solution will be a global solution which admits at zero the convergent series expansion

$$\begin{aligned} \Psi (z)=z^{-\gamma } \sum _{m,n} c_{m} z^m \, g_m, \quad g_m \in V(\lambda ), \end{aligned}$$

for some $\gamma \in \mathbb {C}$. We write the above series more conveniently as

$$\begin{aligned} \Psi (z)(\lambda )=z^{-\gamma } \sum _{m,n} c_{m,n} z^m \lambda ^n, \; c_{m,n} \in V. \end{aligned}$$

(5.9)

Inserting the above Ansatz in the differential equation $\mathcal {L}\psi =0$, the coefficients $c_{m,n}$ are seen to necessarily satisfy the following recurrence

$$\begin{aligned} \left( \mathring{f}+\ell -\gamma +m+k n \right) c_{m,n} + v_\theta c_{m,n-1}+ \sum _{l=1}^m A_l c_{m-l,n}=0 \end{aligned}$$

(5.10)

for some $A_l \in \widetilde{\mathfrak {g}}$. In order to solve the recurrence, it is natural to impose the following two conditions:

• (1) $\gamma$ is an eigenvalue of $\mathring{f}+\ell$ in the representation V;

• (2) $\mathring{f}+\ell -\gamma +m+k n$ is invertible for every $m,n \ge 0, (m,n) \ne (0,0)$.

Regarding condition (1), we notice that in any finite dimensional representation the spectra of $\mathring{f}+\ell$ and of $\ell$ coincide, and that the condition that $\mathring{f}+\ell$ is semisimple is generic in $\ell$. Regarding condition (2), we notice that in any finite dimensional representation V, fixed k, the condition (2) is generic in $\ell$. Therefore, we make the following definition.

Definition 5.2

For $i\in \mathring{I}$ let $P({\tilde{\omega }}_i)$ be the multi-set of weights—with multiplicities—of the fundamental representation $L({\tilde{\omega }}_i)$. Fixed $k \in (0,1)$, $\ell \in \mathring{\mathfrak {h}}$ is said to be generic if $f+\ell$ is semisimple and if for every $i\in \mathring{I}$ and every $\omega \in P({\tilde{\omega }}_i)$ the element

$$\begin{aligned} \mathring{f}+\ell -\omega (\ell )+m+k n \end{aligned}$$

is invertible in ${{\,\textrm{End}\,}}(L({\tilde{\omega }}_i))$ for every $m,n \ge 0, (m,n) \ne (0,0)$.

Proposition 5.3

Fixed k and let $\ell$ be generic as in Definition 5.2. For any $i\in \mathring{I}$ and any $\omega \in P({\tilde{\omega }}_i)$ choose an eigenvector $\chi _{\omega }$ of $f+\ell$ with eigenvalue $\omega (\ell )$.

1. There exists a unique solution $\chi _{\omega } \in \mathcal {A}^{(i)}$, which at 0 admits the Frobenius expansion

$$\begin{aligned} \chi _{\omega }(z,\lambda ) = z^{-\omega (\ell )} \left( \chi _{\omega } + \sum _{(m,n) \ne (0,0)} c_{m,n} z^m \lambda ^n \right) , \end{aligned}$$

(5.11)

convergent in a neighborhood of $z=0$.

2. For every $i\in \mathring{I}$, the collection of all Frobenius solutions $\lbrace \chi _{\omega }(z)\rbrace _{\omega \in P({\tilde{\omega }}_i)}$ is an $\mathcal {O}'$ basis of $\mathcal {A}^{(i)}$.

3. If w in an element of the Weyl group $\mathcal {W}$ of $\mathring{\mathfrak {g}}$, we denote by $\chi ^{(i)}_{w},{\widetilde{\chi }}^{(i)}_{w}\in \mathcal {A}^{(i)}$ the solutions corresponding respectively to the weights $w({\tilde{\omega }}_i)$ and $w({\tilde{\omega }}_i-\alpha _i)$. We can find a normalisation of these solutions such that they satisfy the following set of relations

$$\begin{aligned}&m_i\left( R_i \big (\chi ^{(i)}_{w,-\frac{D_i}{2}}\big ) \wedge {\widetilde{\chi }}^{(i)}_{w,\frac{D_i}{2}} \right) = +e^{D_iw[\alpha _i] (\ell )} \otimes _{j\in \mathring{I}} \otimes _{l=0}^{B_{ij}-1}\chi ^{(j)}_{w,\frac{B_{ij}-1-2l}{2r}} \nonumber \\&m_i\left( R_i \big ( {\widetilde{\chi }}^{(i)}_{w,\frac{-D_i}{2}} \big ) \wedge \chi ^{(i)}_{w,\frac{D_i}{2}} \right) = -e^{-D_iw[\alpha _i] (\ell )} \otimes _{j\in \mathring{I}} \otimes _{l=0}^{B_{ij}-1}\chi ^{(j)}_{w,\frac{B_{ij}-1-2l}{2r}} \end{aligned}$$

(5.12)

where $m_i$ is the linear map defined in (5.8).

Proof

1. In [45, Proposition 5.10], it is proven that the Frobenius series (5.9) and that its analytic continuation belongs to $\mathcal {A}^{(i)}$, in the case $r=1$. The same proof applies to the general case.

The fact that $\chi _{\omega }$ is an eigenvalue of $\mathcal {M}$ is straightforward

$$\begin{aligned} \mathcal {M}(\chi ^{\omega })(z)=e^{-2\pi i \omega (\ell )}z^{-\omega (\ell )} \left( \chi _{\omega } + \sum _{(m,n) \ne (0,0)} c_{m,n} z^m \lambda ^n \right) = e^{-2\pi i \omega (\ell )} \chi ^{\omega }(z) . \end{aligned}$$

(5.13)

2. The Frobenius solutions are a $\mathcal {O}'-$basis of $\mathcal {A}^{(i)}$ since by hypothesis $\lbrace \chi _{\omega }\rbrace _{\omega \in P({\tilde{\omega }}_i)}$ is a basis of $L(\tilde{\omega _i})$.

3. It follows directly from (5.8) and (5.11). $\square$

5.1.2 A basis at infinity. Subdominant solutions

Here we study the asymptotic behaviour at $\infty$ of solutions in $\mathcal {A}^{(i)}$:

• We provide the existence of distinguished bases, with prescribed exponential asymptotic behaviour in sectors of $\widetilde{\mathbb {C}^{*}}$ of amplitude at least $\pi h$, with h the Coxeter number of $\mathfrak {g}$.

• For every such a basis, we select a distinguished solution for its subdominant behaviour, and we show that its asymptotic behaviour holds on a sector of amplitude at least $2 \pi h$.

• We define the central connection matrix and the Stokes matrix, which, under the ODE/IM correspondence, are identified with the Q and T operator-valued functions of the quantum Drinfeld–Sokolov model.

Our results follow rather directly from the following

Theorem 5.4

([17]) Let V a finite dimensional vector space and, for M positive, let $\widetilde{{\mathbb {C}}}^*_M= \lbrace z \in \widetilde{{\mathbb {C}}^*}, |z| > M \rbrace .$ Consider the differential equation

$$\begin{aligned} \psi '(z)= \left( - A \, p(z;\lambda ) + R(z;\lambda ) \right) \psi (z), \quad \psi : \widetilde{{\mathbb {C}}}^*_M \rightarrow V \end{aligned}$$

(5.14)

where M is some positive number, and let

• $A \in End(V)$ be a diagonalisable matrix with eigenvectors $\psi _j$ and eigenvalues $\nu _j$, $j=1,\dots , \dim V$.

• $p(z;\lambda )= z^{\sigma _0}+\sum _{n=1}^{N} c_n(\lambda ) z^{\sigma _n}$ where the exponents $\sigma _n$’s are real and ordered so that

  $$\begin{aligned} \sigma _0>\sigma _1>\dots> \sigma _{N-1}>\sigma _{N} =-1. \end{aligned}$$

  Moreover, the coefficients $c_n(\lambda )$ are analytic bounded functions of the parameter $\lambda$, which belongs to a domain $D \subset {\mathbb {C}}$. Given such a $p(z;\lambda )$ we define its primitive

  $$\begin{aligned} P(z;\lambda )= \frac{z^{\sigma _0+1}}{\sigma _0+1} +\sum _{n=1}^{N-1} c_n(\lambda ) \frac{z^{\sigma _n+1}}{\sigma _n+1}+ c_N \log z. \end{aligned}$$

• $R(z;\lambda )$ be a matrix valued function such that in an arbitrary closed sector of $\widetilde{{\mathbb {C}}}^*_M$ $|R(z;\lambda )|=O(z^{-1-\delta })$ uniformly with respect to $\lambda \in D$.

1. Assume that the interval [a, b] is such that the following condition holds:

$$\begin{aligned} \forall \varphi \in [a,b], \quad {\text {Re}}(\nu _j e^{i\varphi })= {\text {Re}}(\nu _{j'} e^{i\varphi }) \text{ if } \text{ and } \text{ only } \text{ if } \nu _j=\nu _{j'}. \end{aligned}$$

(5.15)

There exists a unique basis of solutions $\psi _j(z;\lambda )$ satisfying the following asymptotics

$$\begin{aligned} \psi _j(z,\lambda )&= \left( \psi _j + O(z^{-\delta }) \right) e^{-\nu _j P(z;\lambda )},\nonumber \\&\text{ as } z \rightarrow \infty \text{ in } \text{ the } \text{ sector } \frac{a}{\sigma _0+1}\le \arg z \le \frac{b+\pi }{\sigma _0+1}, \end{aligned}$$

(5.16)

uniformly with respect to $\lambda \in D$. Moreover, the functions $\psi _j(\cdot ,\cdot )$’s are analytic in $\widetilde{{\mathbb {C}}}^*_M \times D$.

2. Assume that $j_0$ is a subdominant index for the interval [a, b], namely

$$\begin{aligned} \forall \varphi \in [a,b] \text{ and } j \ne j_0, \quad {\text {Re}}(\nu _{j_0} e^{i\varphi }) > {\text {Re}}(\nu _j e^{i\varphi }) . \end{aligned}$$

(5.17)

There exists a unique solution $\Psi (z;\lambda )$, called subdominant, such that

$$\begin{aligned} \Psi (z;\lambda )&= \left( \psi _{j_0} + O(z^{-\delta }) \right) e^{-\nu _{j_0} P(z;\lambda )}, \nonumber \\&\text{ as } z \rightarrow \infty , \text{ in } \text{ the } \text{ sector } \frac{a-\pi }{\sigma _0+1} \le \arg z \le \frac{b + \pi }{\sigma _0+1}, \end{aligned}$$

(5.18)

uniformly with respect to $\lambda \in D$. Moreover, the function $\Psi (\cdot ,\cdot )$ is analytic in $\widetilde{{\mathbb {C}}}^*_M \times D$.

Proof

See [17, Theorem 3.21]. $\square$

Remark 5.5

Note that the interval [a, b] satisfies (5.15) if and only if $[a+m \pi ,b+m \pi ]$ satisfies the same property for every $m \in {\mathbb {Z}}$. Similarly, the interval [a, b] satisfies (5.17) if and only if $[a+2m \pi ,b+2m \pi ]$ satisfies the same property for every $m \in {\mathbb {Z}}$.

Remark 5.6

Theorem 5.4 is an extensions to the complex plane of the classical theorem of Levinson, see [23], about the asymptotic behaviour of solutions to linear ODE on the real axis.

We will also need the definition of a Stokes matrix

Definition 5.7

Let $\psi _j$’s and $\psi '_{j'}$’s be bases of solution of the linear differential equation (5.14), satisfying the asymptotic behaviour (5.16) on the sectors $\Sigma =\lbrace \frac{a}{\sigma _0+1}\le \arg z \le \frac{b+\pi }{\sigma _0+1}\rbrace$ and $\Sigma '=\lbrace \frac{a'}{\sigma _0+1}\le \arg z \le \frac{b'+\pi }{\sigma _0+1}\rbrace$. The bases $\psi _j$’s and $\psi '_j$’s are said to be consecutive bases if $\Sigma \cap \Sigma '= \lbrace \frac{c}{\sigma _0+1} \le \arg z \le \frac{d}{\sigma _0+1} \rbrace$ with $c<d$ such that condition (5.15) holds for all $\varphi \in [c,d]$. The matrix $\mathcal {S}$ of change of basis,

$$\begin{aligned} \psi '_{j'}=\sum _{j=1}^{\dim V} \psi _{j} \mathcal {S}_{j,j'} , \; j' =1,\dots , \dim V, \end{aligned}$$

(5.19)

is called a Stokes matrix.

An important property of a Stokes matrix is that it is unipotent. In fact, we have the following lemma.

Lemma 5.8

With the notation of Definition 5.7. Assume that $\psi _j$’s and $\psi '_{j'}$’s are consecutive bases and the intersection of their respective sectors is $\lbrace \frac{c}{\sigma _0+1} \le \arg z \le \frac{d}{\sigma _0+1} \rbrace$ for some $c<d$. Then

$$\begin{aligned}&\mathcal {S}_{j,j}=1 , \; j=1,\dots , \dim V , \end{aligned}$$

(5.20)

$$\begin{aligned}&\mathcal {S}_{j,j'}=0, \text{ if } j\ne j' \text{ and } Re (\nu _{j} e^{i\varphi }) \le {\text {Re}}(\nu _{j'} e^{i\varphi }), \; \forall \varphi \in [c,d]. \end{aligned}$$

(5.21)

We remark that, due to (5.15), if the condition $Re (\nu _{j} e^{i\varphi }) \le {\text {Re}}(\nu _{j'} e^{i\varphi })$ holds for a $\varphi \in [c,d]$ then it holds for every $\varphi \in [c,d]$.

Proof

By definition of $\Sigma$ and by Theorem 2.5,

$$\begin{aligned} \left( \psi _{j'} + O(z^{-\delta })\right) e^{-\nu _{j'}P(z;\lambda )}=&\sum _{j=1}^{\dim V} \left( \psi _{j} + O(z^{-\delta })\right) e^{-\nu _{j}P(z;\lambda )}\mathcal {S}_{j,j'}, \\&z \rightarrow \infty \text{ and } \frac{c}{\sigma _0+1}\le \arg z \le \frac{d}{\sigma _0+1}. \end{aligned}$$

Fixed a $c>0$ and a $\varphi \in [c,d]$, we let

$$\begin{aligned} l_{\varphi ,c}&= \lbrace z, {\text {Im}}\big (z^{\sigma _0+1}e^{-i\varphi }\big )= c\rbrace \\&= \lbrace |z|^{\sigma _0+1} \sin \big (-\varphi +(\sigma _0+1) \arg z\big ) =c , \; \frac{\varphi }{\sigma _0+1}<\arg z <\frac{\varphi +\pi }{\sigma _0+1} \rbrace . \end{aligned}$$

We parameterise $l_{\varphi ,c}$ by $t \in \mathbb {R}$ in such a way that $z^{\sigma _0+1}\big (l_{\varphi ,c}(t)\big )= (t+ i c) e^{+i\varphi }$. Clearly, $l_{\varphi ,c}$ belongs to the sector $\frac{c}{\sigma _0+1}\le \arg z \le \frac{d}{\sigma _0+1}$ if t is positive and large enough. Hence, restricted to $l_{\varphi ,c}$, the above estimate reads

$$\begin{aligned} \left( \psi _{j'} + O\big (t^{-\delta '}\big )\right) e^{(-\nu _{j'} e^{-i\varphi }) t}= \sum _{j=1}^{\dim V} \mathcal {S}_{j,j'}\left( \psi _{j} + O\big (t^{-\delta '}\big )\right) e^{(-\nu _{j}e^{-i\varphi }) t} , t \rightarrow +\infty , \end{aligned}$$

where $\delta '=\min \lbrace \frac{\delta }{\sigma _0+1}, 1-\frac{\sigma _1+1}{\sigma _0+1}\rbrace$. Multiplying the two sides of the estimate by $e^{\nu _{j'} e^{i\varphi } t}$ and comparing them as $t \rightarrow +\infty$, we obtain the thesis. $\square$

We want to use Theorem 5.4 in order to study asymptotic solutions of global solution $\mathcal {A}^{(i)}$, for $i\in \mathring{I}$. To do that, we consider the differential equation $\mathcal {L}_{\kappa _i}\psi =0$ in the loop realisation

$$\begin{aligned} \mathcal {L}_{\kappa _i}(z;\lambda ) \psi (z;\lambda ) =0, \end{aligned}$$

(5.22)

where

$$\begin{aligned} \psi (\cdot ;\lambda ): \left( \widetilde{\mathbb {C}^*} \setminus \bigcup _{j \in J}\Pi ^{-1} (e^{-2\pi i \kappa _i} w_j)\right) \rightarrow L({\tilde{\omega }}_i). \end{aligned}$$

Explicitly, the operators $\mathcal {L}_{\kappa _i}(z;\lambda )$ are given by

$$\begin{aligned} \mathcal {L}_{\kappa _i}(z;\lambda )&= \partial _z+ \frac{1}{z}\Biggl (\mathring{f}+ e^{2 \pi i \kappa _i}z\big ( 1+ e^{-2 \pi i \kappa _i}\lambda z^{k-1}\big )v_\theta + \ell + \nonumber \\&\quad +\sum _{j =1}^N \frac{ z }{ z-w}\left( -\theta ^\vee + X(j) \right) \Biggl ), \end{aligned}$$

(5.23)

and the coefficients $\kappa _i$’s are as in Table 3. Notice that we cannot apply Theorem 5.4 to the differential equation (5.22), as it is not of the form (5.14). However, we will find that (5.22) is Gauge equivalent to an equation of the form (5.14). More precisely, we fill find a gauge G such that

$$\begin{aligned} G. \mathcal {L}_{\kappa _i}(z;\lambda ) =\partial _z+ q_i(z;\lambda ) \Lambda (\kappa _i)+ o(z^{-1}), \end{aligned}$$

(5.24)

where $\Lambda (\kappa _i)$ is the cyclic element $\mathring{f} + e^{2 \pi i \kappa _i} v_{\theta }$ introduced in (2.34), while

$$\begin{aligned} q_i(z;\lambda )= z^{\frac{1}{h}} \left( 1 + \sum _{l=1}^{\lfloor \frac{1}{(1-k)h} \rfloor } c_l \, e^{-2 l \pi i \kappa _i }\lambda ^l z^{l(1-k)}\right) .\end{aligned}$$

Here $c_l$ are the coefficients of the McLaurin expansion of $(1-w)^{\frac{1}{h}}$.

Before proceeding further, we recall some facts about the spectrum of $\Lambda (\kappa _i)$. The cyclic element is regular and semisimple [36], and its centralizer is a Cartan subalgebra, say $\underline{\tilde{\mathfrak {h}}}$ (in the case $r=1$, $\underline{\tilde{\mathfrak {h}}}$ is said to be in apposition with respect to $\tilde{\mathfrak {h}}$ [12, 38]). Letting $P(\underline{{\tilde{\omega }}_i})$ be the multi-set of weights of the representation $L({\tilde{\omega }}_i)$, corresponding to the Cartan sub-algebra $\underline{\tilde{\mathfrak {h}}}$, the eigenvalues and eigenvectors are denoted by $\mu ^{(i)}_{{\underline{\omega }}}={\underline{\omega }} \left( \Lambda (\kappa _i) \right)$ and $\psi ^{(i)}_{{\underline{\omega }}}$. In particular, the maximal eigenvalue $\mu ^{(i)}$ studied in Theorem 2.5 corresponds to the fundamental weight $\underline{{\tilde{\omega }}_i}$, namely $\mu ^{(i)}=\mu ^{(i)}_{\underline{{\tilde{\omega }}_i}}$ and $\psi ^{(i)}=\psi ^{(i)}_{\underline{{\tilde{\omega }}_i}}$; see [46, 47].

Definition 5.9

We say that [a, b] is a good interval for $\Lambda (\kappa _i)$ if for every $\varphi \in [a,b]$:

$$\begin{aligned} {\text {Re}}(\mu ^{(i)}_{{\underline{\omega }}} e^{i\varphi }) = {\text {Re}}(\mu ^{(i)}_{{\underline{\omega }}'}e^{i\varphi }), \text{ if } \text{ and } \text{ only } \text{ if } \mu ^{(i)}_{{\underline{\omega }}}=\mu ^{(i)}_{{\underline{\omega }}'}. \end{aligned}$$

(5.25)

We let $\zeta ^{(i)}$ be the supremum of all positive numbers $\zeta$⁴ such that for every $\varphi \in [-\zeta ,\zeta ]$ and ${\underline{\omega }}'\ne \underline{{\tilde{\omega }}_i}$:

$$\begin{aligned} {\text {Re}}(\mu ^{(i)}_{\underline{{\tilde{\omega }}_i}} e^{i\varphi }) > {\text {Re}}(\mu ^{(i)}_{{\underline{\omega }}} e^{i\varphi }). \end{aligned}$$

(5.26)

Proposition 5.10

Let $i \in i\in \mathring{I}$ and $S_i$ be the primitive of $z^{-1}q_i(z,\lambda )$ given by

$$\begin{aligned} S_i(z;\lambda )= {\left\{ \begin{array}{ll} &{} h \, z^{\frac{1}{h}} \left( 1 + {\sum }_{l=1}^{\lfloor \frac{1}{(1-k)h} \rfloor } \frac{c_l e^{-2 l \pi i \kappa _i }\lambda ^l z^{-l(k-1)} }{1- h l (1-k)} \right) , \quad \frac{1}{h(1-k)} \notin \mathbb {N}\\ &{} h \, z^{\frac{1}{h}} \left( 1 + {\sum }_{l=1}^{m-1} \frac{c_l e^{2 l \pi i \kappa _i }\lambda ^l z^{-l(k-1)}}{1- l\,m} \right) + c_{m}\log z , \, m:=\frac{1}{h(1-k)} \in \mathbb {N} \end{array}\right. } . \end{aligned}$$

(5.27)

There exists a $\delta >0$ such that:

1. If [a, b] a good interval for $\Lambda (\kappa _i)$, for every $\lambda \in \mathbb {C}$ the differential equation (5.23) admits a unique basis of solutions ${\widetilde{\Phi }}_{{\underline{\omega }}}^{(i)}(\cdot ;\lambda )$, with ${\underline{\omega }} \in P(\underline{{\tilde{\omega }}_i})$, such that

$$\begin{aligned} {\widetilde{\Phi }}_{{\underline{\omega }}}^{(i)}(z;\lambda )= z^{\frac{1}{h}{{\,\textrm{ad}\,}}\mathring{\rho }^\vee } \left( \psi _{{\underline{\omega }}}^{(i)} + O\big (z^{-\delta }\big ) \right) e^{-\mu _{{\underline{\omega }}}^{(i)} S_i(z;\lambda )}, \end{aligned}$$

(5.28)

as $z \rightarrow + \infty$ in the closed sector $a h \le \arg z \le h (\pi + b)$. Moreover, the solutions ${\widetilde{\Psi }}_{{\underline{\omega }}}^{(i)}$’s are entire functions of the parameter $\lambda$. Therefore, the elements $\Phi ^{(i)}_{{\underline{\omega }}} \in \mathcal {A}^{(i)}, {\underline{\omega }} \in P(\underline{{\tilde{\omega }}_i})$ corresponding to ${\widetilde{\Psi }}^{(i)}_{{\underline{\omega }}}(\cdot ;\lambda )$ under the isomorphism $z^{k\textbf{d}}$, i.e. $\Phi ^{(i)}_{{\underline{\omega }}} (z)(\lambda )={\widetilde{\Phi }}^{(i)}_{{\underline{\omega }}}(z;\lambda z^{k})$, form a $\mathcal {O}'$-basis of $\mathcal {A}^{(i)}$.

2. For every $\lambda \in \mathbb {C}$, the differential equation (5.23) admits a unique solution ${\widetilde{\Phi }}^{(i)}(\cdot ;\lambda )$ such that for all $\theta < \theta ^{(i)}$,

$$\begin{aligned} {\widetilde{\Phi }}^{(i)}(z;\lambda )= z^{\frac{1}{h}{{\,\textrm{ad}\,}}\mathring{\rho }^\vee } \left( \psi ^{(i)} + O\big (z^{-\delta }\big ) \right) e^{-\mu ^{(i)} S_i(z;\lambda )}, \end{aligned}$$

(5.29)

as $z \rightarrow + \infty$ in the closed sector $|\arg z|\le h (\pi +\theta ^{(i)})$. Moreover, the solution ${\widetilde{\Phi }}^{(i)}$ is an entire function of the parameter $\lambda$. We denote by $\Psi ^{(i)}$ the element of $\mathcal {A}^{(i)}$ defined by $\Phi ^{(i)}(z;\lambda )= {\widetilde{\Phi }}^{(i)}(z;\lambda z^{k})$.

Proof

We find a Gauge transformation $G=G_2 \circ G_1$ that transforms the connection $\mathcal {L}_{\kappa _i}(z;\lambda )$ to

$$\begin{aligned} \partial _z+ q_i(z;\lambda ) \Lambda (\kappa _i)+ O(z^{-1-\delta }), \text{ for } \text{ some } \delta >0. \end{aligned}$$

By definition of the function $q_i$, we have that

$$\begin{aligned} z^{\frac{1}{h}}\big ((1+ e^{-2 \pi i \kappa _i} \lambda z^{k-1})\big )^{\frac{1}{h}}-q_i(z;\lambda )= O\left( z^{-\delta _0} \right) , \end{aligned}$$

(5.30)

with

$$\begin{aligned} \delta _0=(1-k)\left( \lfloor \frac{1}{(1-k)h} \rfloor +1- \frac{1}{(1-k)h}\right) >0. \end{aligned}$$

Therefore acting with $G_1=\big (q(z;\lambda )\big )^{-{{\,\textrm{ad}\,}}\mathring{\rho }^\vee }$, we obtain

$$\begin{aligned} G_1 \mathcal {L}_{\kappa _i}(z;\lambda )=\partial _z+ z^{-1}q_i(z;\lambda ) \Lambda (\kappa _i) + \frac{\ell + \frac{1}{h^\vee } \mathring{\rho }^\vee - r |J| r \theta ^\vee }{z}+ O(z^{-1-\delta _0}). \end{aligned}$$

(5.31)

Now, we let ${\tilde{n}}$ be the unique element in $\langle e_1,\dots ,e_n \rangle$ such that $[{\tilde{n}},\mathring{f}]=\ell + \frac{1}{h^\vee } \mathring{\rho }^\vee - N r \theta ^\vee$. Acting with $G_2=\exp \left( q_i(z;\lambda )^{-1} {\tilde{n}}\right)$, we get

$$\begin{aligned} \mathcal {L}_{\kappa _i,*}(z;\lambda ):= G_2 \circ G_1 \mathcal {L}_{\kappa _i}(z;\lambda )=\partial _z+ z^{-1} q_i(z;\lambda ) \Lambda (\kappa _i) + O(z^{-1-\delta }),\end{aligned}$$

with $\delta = \min \lbrace \frac{1}{h},\delta _0\rbrace >0$.

Applying Theorem 5.4 to $\mathcal {L}_{\kappa _i,*}(z;\lambda )$, we deduce the thesis. $\square$

5.2 Central connection matrix and Stokes matrix. Q and T functions

Let $\zeta _*>0$ be a sufficiently small numbers so that for every $\zeta , 0<\zeta <\zeta _*$, the interval $[-\zeta _*,-\zeta ]$ is a good interval for $\Lambda (\kappa _i)$ for every $i \in \mathring{I}$. After Proposition 5.10, $\mathcal {A}^{(i)}$ admits the $\mathcal {O}'$-basis $\Psi ^{(i)}_{{\underline{\omega }}}, {\underline{\omega }} \in P({\underline{\omega }}_i)$, where $\Psi ^{(i)}_{{\underline{\omega }}}$ is such that ${\widetilde{\Psi }}^{(i)}_{{\underline{\omega }}}= z^{-k \textbf{d}} \Psi ^{(i)}_{{\underline{\omega }}}$ satisfies the asymptotics (5.28) in the sector $[-\zeta _*,\pi -\zeta ]$ for all $0<\zeta <\zeta _*$. Moreover, the solution $\Psi ^{(i)}:=\Psi ^{(i)}_{\underline{{\tilde{\omega }}_i}}$ is subdominant along the real positive axis, and admits the asymptotic behaviour (5.29) on the larger sector $[-\zeta _*-\pi ,\zeta _*+\pi ]$.

We also know from Proposition 5.3, that—assuming that $\ell$ is generic—$\mathcal {A}^{(i)}$ admits another $\mathcal {O}'$ distinguished basis—the eigenbasis of the monodromy operator $\mathcal {M}$—whose elements are the Frobenius solutions $\chi _{\omega }$, $\omega \in P({\tilde{\omega }}_i)$.

The two bases are therefore related by invertible matrix $Q^{(i)}$, called central connection matrix, whose coefficients belongs to $\mathcal {O}'$. We write

$$\begin{aligned} \Psi ^{(i)}_{{\underline{\omega }}}= \sum _{\omega \in P({\tilde{\omega }}_i)} Q^{(i)}_{\omega ,{\underline{\omega }}} \chi _{\omega }, \qquad {\underline{\omega }} \in P(\underline{{\tilde{\omega }}_i}). \end{aligned}$$

(5.32)

In particular, denoting by $Q^{(i)}_{\omega }:= Q^{(i)}_{\omega ,\underline{{\tilde{\omega }}_i}}$, we have that

$$\begin{aligned} \Psi ^{(i)}= \sum _{\omega \in P({\tilde{\omega }}_i)} Q^{(i)}_{\omega } \chi _{\omega }(z;\lambda ). \end{aligned}$$

(5.33)

The coefficients $Q^{(i)}_{\omega ,\omega '} \in \mathcal {O}'$ or $Q^{(i)}_{\omega } \in \mathcal {O}'$ are called Q functions.

We define now the Stokes matrix $\mathcal {T}$. Let us a fix a $0<{\widetilde{\zeta }}_*<\zeta$. It is straightforward to check that the interval $[-\zeta ,-{\widetilde{\zeta }}]$ is a good interval for $\Lambda (\kappa _i)$ for every i. Therefore, there exists a $\mathcal {O}'$ basis of $\mathcal {A}^{(i)}$, $\Xi ^{(i)}_{\omega }, \omega \in P(\underline{{\tilde{\omega }}_i})$, such that ${\widetilde{\Xi }}^{(i)}_{\omega }=z^{-k \textbf{d}} \Xi ^{(i)}_{\omega }$ has asymptotic behaviour (5.29) in the sector $[-\zeta _*-\pi ,-\widetilde{\zeta _*}]$. Moreover, the bases ${\widetilde{\Psi }}^{(i)}_{{\underline{\omega }}}$’s—which was defined above—and ${\widetilde{\Xi }}^{(i)}_{{\underline{\omega }}'}$’s are consecutive bases of (5.23), according to the Definition 5.7 above.

We let therefore $\mathcal {T}^{(i)}$ be the matrix of change of basis, with coefficients in $\mathcal {O}'$, defined by

$$\begin{aligned} \Psi ^{(i)}_{{\underline{\omega }}'}= \sum _{{\underline{\omega }} \in P({\tilde{\omega }}_i)} \Xi ^{(i)}_{{\underline{\omega }}} \mathcal {T}^{(i)}_{{\underline{\omega }},{\underline{\omega }}'}, \; {\underline{\omega }}' \in P({\tilde{\omega }}_i). \end{aligned}$$

(5.34)

The coefficients $\mathcal {T}^{(i)}_{{\underline{\omega }},{\underline{\omega }}'}$ are known as $\mathcal {T}$ functions.

Since the bases ${\widetilde{\Psi }}^{(i)}_{{\underline{\omega }}}$’s and ${\widetilde{\Xi }}^{(i)}_{{\underline{\omega }}'}$’s $\mathcal {T}^{(i)}$ are consecutive then $\mathcal {T}^{(i)}$ is Stokes matrix. The following is direct corollary of Lemma 5.8

$$\begin{aligned}&\mathcal {T}^{(i)}_{{\underline{\omega }},{\underline{\omega }}}=1, \forall {\underline{\omega }} \in P(\underline{{\tilde{\omega }}_i}),\end{aligned}$$

(5.35)

$$\begin{aligned}&\mathcal {T}^{(i)}_{{\underline{\omega }},{\underline{\omega }}'}=0, \text{ if } {\underline{\omega }} \ne {\underline{\omega }}' \text{ and } {\text {Re}}\big ( e^{i\varphi }\mu _{{\underline{\omega }}}\big ) \le {\text {Re}}\big (e^{i\varphi } \mu _{{\underline{\omega }}'} \big ) \text{ for } \varphi \in [-\zeta _*,-\widetilde{\zeta _*}]. \end{aligned}$$

(5.36)

Remark 5.11

As we have already stated in the introduction, the definition of $Q^{(i)}$ matrix and of the Stokes matrix $\mathcal {T}^{(i)}$ is, for the general Lie algebra, a novelty of the present work. The thorough study of the matrices $\mathcal {T}^{(i)}$’s and $Q^{(i)}$’s is beyond the scope of the present paper. In particular, we expect but we do not discuss here that the matrices $\mathcal {T}^{(i)}$’s and $Q^{(i)}$ satisfy the $\mathcal {T}Q$ relations described in [29], which generalize the classical Baxter’s $\mathcal {T}Q$ relations. We only note here that in the case $\tilde{\mathfrak {g}}=A_n$ we have $\Psi ^{(1)}= \mathcal {M}^{h-1} \Xi ^{(1)}$, where $\mathcal {M}$ is the monodromy operator. The $\mathcal {T}Q$ relations for the algebra $A_n$ with $r=1$ follow from this; see [19] for details.

5.3 $\Psi$-system, QQ system and the Bethe Equations

Let $\Psi ^{(i)}\in \mathcal {A}^{(i)}$, $i \in \mathring{I}$ be the elements defined in Proposition 5.12. They satisfy the following system of nonlinear relations known as $\Psi$-system

Proposition 5.12

For every $i\in \mathring{I}$, let $\Psi ^{(i)}\in \mathcal {A}^{(i)}$ be defined by

$$\begin{aligned} \Psi ^{(i)}(z)(\lambda )={\widetilde{\Psi }}^{(i)}(z;\lambda z^{-k}). \end{aligned}$$

Then the following identity, known as $\psi -$system holds true:

$$\begin{aligned} m_i \left( R_i \left( \Psi ^{(i)}_{-\frac{D_i}{2}}\right) \wedge \Psi ^{(i)}_{\frac{D_i}{2}} \right) = \bigotimes _{j \in \mathring{I}} \bigotimes _{l=0}^{B_{ij}-1} \Psi ^{(j)}_{\frac{B_{ij}-1-2l}{r}}, \end{aligned}$$

(5.37)

where $m_i$ is the linear map defined in (5.8).

Proof

This is proven in [47, Theorem 4.7 (v)] for the ground state oper. The same proofs hold for all FFH opers, since it only depends on the asymptotic behaviour (5.29), which, as Proposition 5.10, is independent of the additional singularities. $\square$

As a corollary, we have the following

Theorem 5.13

Assume that the pair $(k,\ell )$ is generic as in Definition 5.2, and that $\mathring{f}+\ell$ is semisimple. Fixing an element w of the Weyl group of $\mathring{\mathfrak {g}}$, denote $Q_w^{(i)}=Q_{w({\tilde{\omega }}_i)}$ and ${\widetilde{Q}}^{(i)}_w=Q_{w({\tilde{\omega }}_i-\alpha _i)}$.

1. The following system, known as $QQ-$system, holds for all $i \in \mathring{I}$ and all $\lambda \in \mathbb {C}$

$$\begin{aligned} \begin{aligned} \prod _{j\in \mathring{I}}\prod _{s=0}^{B_{ij}-1}Q_w^{(j)}(q^{\frac{B_{ij}-1-2s}{r}} \lambda )&=e^{\pi i D_i\langle \ell ,w({\tilde{\alpha }}_i)\rangle }Q_w^{(i)}(q^{D_i} \lambda ){\widetilde{Q}}_w^{(i)}(q^{- D_i}\lambda )\\&-e^{- \pi i D_i\langle \ell ,w({\tilde{\alpha }}_i)\rangle }Q_w^{(i)}(q^{-D_i }\lambda ){\widetilde{Q}}_w^{(i)}(q^{D_i}\lambda ) \,, \end{aligned} \end{aligned}$$

(5.38)

where $q=e^{\pi i k}$.

2. Let $\lambda ^*$ be a zero of $Q_w^{(i)}$ such that

$$\begin{aligned} \prod _{j\in \mathring{I}}\prod _{\ell =0}^{B_{ij}-1}Q^{(j)}(q^{\frac{B_{ij}-1-2\ell }{r}-D_i} \lambda ^*)\ne 0. \end{aligned}$$

Then, the following Bethe Equations hold

$$\begin{aligned} \prod _{j\in \mathring{I}}e^{i\pi {\overline{C}}_{sj}\theta ^j_w}\frac{Q^{(j)}(q^{{\overline{C}}_{sj}}\lambda ^*)}{Q^{(j)}(q^{-{\overline{C}}_{sj}}\lambda ^*)}=-1, \end{aligned}$$

(5.39)

where $\theta ^j_w=\langle \ell ,w({\tilde{\omega }}_j)\rangle$, and ${\overline{C}}_{ij}=C_{ji}D_i$ is the symmetrized Cartan matrix of $^L\mathring{\mathfrak {g}}$.

Proof

Plug the decomposition (5.33) into the $\Psi$-system (5.37) and use the relation (5.12). $\square$

5.4 Simplifications in the case r=1

For the benefit of the reader we specialise the $\Psi$-system and the QQ system to the untwisted case $r=1$. Recall that in this case the algebras $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$ coincide, as well as the index sets $\mathring{I}={\tilde{I}}=\{1,\dots ,n\}$. The following reductions are straightforward to check—see also [45].

The linear maps $m_i$ defined in equation (5.8) read

$$\begin{aligned} m_i:\bigwedge ^2 \mathcal {A}^{(i)}_{\frac{1}{2}}\longrightarrow \bigotimes _{j\in \mathring{I}} \mathcal {A}^{(j)}_{B_{ij}}\,, \qquad i\in \mathring{I}, \end{aligned}$$

(5.40)

so that the $\Psi$-system takes the form

$$\begin{aligned} m_i \left( \Psi ^{(i)}_{-\frac{1}{2}} \wedge \Psi ^{(i)}_{\frac{1}{2}} \right) = \bigotimes _{j \in \mathring{I}} \Psi ^{(j)}_{B_{ij}}. \end{aligned}$$

(5.41)

Similarly, the QQ-system reduces to

$$\begin{aligned} \begin{aligned} \prod _{j\in \mathring{I}}Q_w^{(j)}( \lambda )&=e^{\pi \sqrt{-1}\langle \ell ,w({\tilde{\alpha }}_i)\rangle }Q_w^{(i)}(q \lambda ){\widetilde{Q}}_w^{(i)}(q^{- 1}\lambda )\\&\quad -e^{- \pi \sqrt{-1}\langle \ell ,w({\tilde{\alpha }}_i)\rangle }Q_w^{(i)}(q^{-1 }\lambda ){\widetilde{Q}}_w^{(i)}(q\lambda ) \,, \end{aligned} \end{aligned}$$

(5.42)

where $q=e^{\pi \sqrt{-1} k}$, and the Bethe Equations become

$$\begin{aligned} \prod _{j\in \mathring{I}}e^{i\pi C_{sj}\theta ^j_w}\frac{Q^{(j)}(q^{C_{sj}}\lambda ^*)}{Q^{(j)}(q^{-C_{sj}}\lambda ^*)}=-1, \qquad \theta ^j_w=\langle \ell ,w({\tilde{\omega }}_j)\rangle , \end{aligned}$$

(5.43)

where $\lambda ^*$ is any zero of $Q_w^{(i)}$ such that $\prod _{j\in \mathring{I}}Q^{(j)}( \lambda ^*)\ne 0$.

6 Feigin–Frenkel–Hernandez $\mathfrak {g}$-Opers

In this section we define Feigin–Frenkel–Hernandez $\mathfrak {g}$-opers (or FFH $\mathfrak {g}$-opers) as Gauge equivalence classes of connections satisfying certain local assumptions, to be specified below. This is essentially the approach originally given in [30]. The main result is Theorem 6.6, where we prove that each FFH $\mathfrak {g}$-oper admits an (essentially unique) representative as a FFH connection (3.1). In the Appendix, we define the notion of affine twisted parabolic Miura $\mathfrak {g}$-opers, and prove that FFH $\mathfrak {g}$-opers belong to this class of opers.

6.1 A class of affine $\mathfrak {g}$-opers

Let $\mathcal {K}$ be the field of meromorphic functions on $\mathbb {C}^*$, and for any vector space V denote by $V(\mathcal {K})$ the space of meromorphic functions from $\mathbb {C}^*$ to V.

Definition 6.1

We denote by ${{\,\textrm{op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$ the set of meromorphic operators of the form

$$\begin{aligned} L(z)=\partial _z+\frac{1}{z}\left( f+\xi (z^r)\textbf{d}+b(z^r)+zc(z^r)v_\theta \right) , \end{aligned}$$

(6.1)

with $\xi \in \mathcal {K}$ and $b\in \mathring{\mathfrak {b}}^+(\mathcal {K})$, and $v_\theta \in \tilde{\mathfrak {g}}$ is the element introduced in Definition 2.1.

It is clear that FFH connections (3.1) belong to this class. We consider the Gauge group

$$\begin{aligned} \mathring{\mathcal {N}}^{\sigma }(\mathcal {K})=\{\exp (y(z^r))\,|\,y\in \mathring{\mathfrak {n}}^+(\mathcal {K})\}, \end{aligned}$$

(6.2)

and we define its action on the operators (6.1) as follows. Since $f=\mathring{f}+f_0$, every connection (6.1) can be written as $L(z)=\mathring{L}(z)+(f_0-k\textbf{d}+zv_\theta )/z$, where

$$\begin{aligned} \mathring{L}(z)=\partial _z+\frac{1}{z}\left( \mathring{f}+b(z^r)\right) \end{aligned}$$

and $b\in \mathring{\mathfrak {b}}^+(\mathcal {K})$. Then we define

$$\begin{aligned} \exp ({{\,\textrm{ad}\,}}y(z^r))\mathring{L}(z)=\partial _z+\frac{1}{z}\left( \sum _{k\ge 0}\frac{({{\,\textrm{ad}\,}}y)^k}{k!}\left( f+b(z)\right) -rz^{r}\sum _{k\ge 0}\frac{({{\,\textrm{ad}\,}}y)^k}{(k+1)!}\frac{dy}{dz}\right) , \end{aligned}$$

(6.3)

and

$$\begin{aligned} \exp ({{\,\textrm{ad}\,}}y(z^r))L(z)=\exp ({{\,\textrm{ad}\,}}y(z^r))\mathring{L}(z)+\frac{1}{z}(f_0-k\textbf{d}+zv_\theta ). \end{aligned}$$

(6.4)

The latter definition is consistent with the fact that $f_0$, $\textbf{d}$ and $v_\theta$ are invariant under the action of the unipotent group $\mathring{\mathcal {N}}$. Note that the class of operators (6.1) is invariant under the action $\mathring{\mathcal {N}}(\mathcal {K})$.

Definition 6.2

We define ${{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})={{\,\textrm{op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})/\mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$, and we denote by [L] the equivalence class of $L\in {{\,\textrm{op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$.

It is a result of Kostant [38] that the Borel subalgebra $\mathring{\mathfrak {b}}^+=\mathring{\mathfrak {h}}\oplus \mathring{\mathfrak {n}}^+$ of $\mathring{\mathfrak {g}}$ admits the decomposition

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

(6.5)

where $\mathfrak {s}$ is a ${{\,\textrm{ad}\,}}_{\mathring{\rho }^\vee }-$invariant subspace, known as transversal subspace, of dimension $n={{\,\textrm{rank}\,}}{\mathring{\mathfrak {g}}}$, and there exists an isomorphism of affine varieties

$$\begin{aligned} \mathring{\mathcal {N}}\times (\mathring{f}+\mathfrak {s})\longrightarrow \mathring{f}+\mathring{\mathfrak {b}}^+. \end{aligned}$$

(6.6)

The choice of $\mathfrak {s}$ is not unique, and we will consider from now on a fixed but otherwise arbitrary transversal subspace.

Definition 6.3

Let $\mathfrak {s}\subset \mathring{\mathfrak {b}}^+$ be a transversal subspace, as in (6.5). The operator (6.1) is said to be in canonical form (with respect to $\mathfrak {s}$) if $b\in \mathfrak {s}(\mathcal {K})$.

Proposition 6.4

Fixed a transversal subspace $\mathfrak {s}\subset \mathring{\mathfrak {b}}^+$, each oper $[L(z)]\in {{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$ admits a unique canonical form (with respect to $\mathfrak {s}$).

Proof

Since $v_\theta$ and $\textbf{d}$ are $\mathring{\mathcal {N}}$-invariant, it is sufficient to prove the proposition for an operator of the form $L(z;\lambda )=\partial _z+\left( \mathring{f}+b(z^r)\right) /z$. For these operators the result is well-known [22]. $\square$

6.2 FFH $\mathfrak {g}$-opers

We can now define FFH $\mathfrak {g}$-opers. Due to (6.6), for every $\ell \in \mathring{\mathfrak {h}}$ there exists a unique pair $(N,{\bar{\ell }})\in \mathring{\mathcal {N}}\times \mathfrak {s}$ such that $N(\mathring{f}+\ell )=\mathring{f}+{\bar{\ell }}$. It follows that in the case $J= \emptyset$ the FFH connection (3.1) has a canonical form

$$\begin{aligned} \mathcal {L}_{G,\mathfrak {s}}(z)= \partial _z+ \frac{1}{z}\left( f+{\bar{\ell }} -k \, \textbf{d}+ zv_\theta \right) . \end{aligned}$$

(6.7)

Definition 6.5

Fixed ${\bar{\ell }} \in {\mathfrak {s}}$ and $k \in (0,1)$, a FFH $\mathfrak {g}$-oper is an element of ${{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$ whose canonical form is given by

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

(6.8)

where $s\in \mathfrak {s}(\mathcal {K})$ satisfies four assumptions:

Assumption 1

The (dominant term of the) asymptotic behaviour of solutions in a neighbourhood of 0 is independent on s:

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

Assumption 2

The (dominant term of the) asymptotic behaviour of solutions in a neighbourhood of $\infty$ is independent on s:

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

Assumption 3

If $w \in {\mathbb {C}}^*$ is a singularity of s then locally

$$\begin{aligned} s(z^r)=\sum _{i=1}^n \sum _{l\ge 0}^{d_i} \frac{zs^{d_i}_l}{(z-w)^{d_i+1-l}}, \end{aligned}$$

where $s_l^{d_i}\in \mathfrak {s}\cap \mathfrak {g}^{d_i}$, $l=0,\dots ,d_i$, and where ${\bar{s}}=\sum _i s_0^{d_i} \in {\mathfrak {s}}$ is the unique element in ${\mathfrak {s}}$ such that the $\mathring{f}/w-\mathring{\rho }^\vee +{\bar{s}}$ and $\mathring{f}/w-\mathring{\rho }^\vee -\theta ^\vee \in \mathring{\mathfrak {g}}$ are conjugated. In other words, every additional singularity is a regular singularity and close to the singularity the oper is locally Gauge equivalent to

$$\begin{aligned} \partial _z+ \frac{\mathring{f}/w-\mathring{\rho }^\vee -\theta ^\vee }{z-w}+O(1). \end{aligned}$$

Assumption 4

If $w \in {\mathbb {C}}^*$ is a singularity of s, the monodromy at w is trivial for every $\lambda \in \mathbb {C}$.

The above definition was given in [25] in the case $r=1$ and in [30] in the general case.

Theorem 6.6

Any FFH $\mathfrak {g}$-oper admits a representative of the form (3.1) for some $J \subset \mathbb {N}$, with $(w_j,X(j),y(j)) \in {\mathbb {C}}^*\times \mathring{\mathfrak {n}}^+ \times \mathbb {C}$, $j\in J$. Moreover, the above representative is essentially unique: fixed $\ell \in \mathring{\mathfrak {h}}$ such that $f+{\bar{\ell }}$ and $f+\ell \in \mathring{\mathfrak {g}}$ are conjugated, two such opers coincide if and only if their representatives (3.1) coincide.

Proof

Recall that $f=\mathring{f}+f_0$. Since the elements $\textbf{d}$, $f_0$ and $v_\theta$ are invariant under the action of the Gauge group $\mathring{\mathcal {N}}(\mathcal {K})$, the thesis is equivalent to the existence and uniqueness of a representation of the form

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

(6.9)

for a $\mathring{\mathfrak {g}}$-oper

$$\begin{aligned} \mathring{\mathcal {L}}= \partial _z+ \frac{1}{z}\left( \mathring{f}+b(z)\right) , \end{aligned}$$

(6.10)

with $b \in \mathring{\mathfrak {b}}^+(\mathcal {K})$ and $b(z)=b(\varepsilon z)$, whose canonical form satisfies Assumptions 1,2,3. We prove it as follows. Due to [45, Proposition 4.7], the canonical form of an oper (6.10) (not necessarily $\sigma -$invariant) which satisfies Assumptions 1,2, and 3, is

$$\begin{aligned} \mathring{\mathcal {L}}_{\mathfrak {s}}= \partial _z+ \frac{\mathring{f}+{\bar{\ell }} }{z} + \sum _{j =1}^M \sum _{i=1}^n \sum _{l=0}^{d_i} \frac{s^{d_i}_l(j)}{(z-{\tilde{w}}_j)^{d_i+1-l}}, \end{aligned}$$

(6.11)

for some $M \in \mathbb {N}$, ${\tilde{w}}_j \in \mathbb {C}^*$, and $s^{d_i}_l(j) \in \mathfrak {s}\cap \mathring{\mathfrak {g}}^{d_i}$, where ${\bar{\ell }} \in {\mathfrak {s}}$ is the unique element such that the $\mathring{f}+{\bar{\ell }}$ and $\mathring{f}+\ell \in \mathring{\mathfrak {g}}$ are conjugated, and for $j=1,\dots ,M$ the element ${\bar{s}}(j)=\sum _i s^{d_i}(j) \in {\mathfrak {s}}$ is the unique element such that $\mathring{f}/{\tilde{w}}_j-\mathring{\rho }^\vee +{\bar{s}}(j)$ and $\mathring{f}/{\tilde{w}}_j-\mathring{\rho }^\vee -\theta ^\vee \in \mathring{\mathfrak {g}}$ are conjugated.

In addition, according to [45, Theorem 6.1], if we fix $\ell$, an oper of the form (6.11) can uniquely be represented as

$$\begin{aligned} \mathring{\mathcal {L}}&= \partial _z+ \frac{\mathring{f}+\ell }{z} + \sum _{k =1}^M \frac{1}{z-{\tilde{w}}_j} \left( -\theta ^\vee + {\tilde{X}}(j) \right) , \end{aligned}$$

(6.12)

for some ${\tilde{X}}(j) \in \mathring{\mathfrak {g}}$ via a Gauge transform belonging to $\mathring{\mathcal {N}}(\mathcal {K})$. Imposing the $\sigma -$invariance it follows that the set of additional singularities must be invariant under rotations by $\varepsilon$ (namely that ${\tilde{w}}_j$ is an additional singularity if and only if $\varepsilon {\tilde{w}}_j$ is another additional singularity) and that (6.12) has to be of the form (6.9), with $|J|=M/3$ and for certain, uniquely determined, coefficients X(j), $j\in J$. $\square$

Affine Twisted Parabolic Miura Opers

We introduce an untwisted affine Kac-Moody algebra $\bar{\mathfrak {g}}$ and we realize the algebras $\mathfrak {g}$, $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$ introduced in Sect. 2 as subalgebras of $\bar{\mathfrak {g}}$. The algebra $\bar{\mathfrak {g}}$ admits a Dynkin diagram automorphism which we denote as $\sigma$, which is the same letter we used for the Dynkin diagram automorphism of $\tilde{\mathfrak {g}}$ considered in Sect. 2. The restriction of $\sigma \in {{\,\textrm{Aut}\,}}{\bar{\mathfrak {g}}}$ to the subalgebra $\tilde{\mathfrak {g}}$ coincides with Dynkin diagram automorphism previously defined. We use this construction to introduce a class of opers which we call affine twisted parabolic Miura opers. After describing their properties, we show that the FFH connections considered in (3.1) are representatives of this class, thus proving a conjecture by Hernandez and Frenkel [30].

Schematically, the Lie algebras $\bar{\mathfrak {g}}, \tilde{\mathfrak {g}},\mathfrak {g}$ and $\mathring{\mathfrak {g}}$ are related as follows.

• The algebra $\bar{\mathfrak {g}}$ is one of the untwisted affine Kac-Moody algebra listed in Table 4. We index the nodes of the Dynkin diagram of $\bar{\mathfrak {g}}$ by the set

  $$\begin{aligned} {\bar{I}}=\{0,1,\dots ,{\bar{n}}\}. \end{aligned}$$

  The algebra $\bar{\mathfrak {g}}$ admits a Dynkin diagram automorphism $\sigma$, of order r. The trivial ($r=1$) cases are shown in Table , the non-trivial ($r>1$) cases in Table .

• The Lie algebra $\tilde{\mathfrak {g}}$ is the simple finite dimensional Lie algebra introduced in Sect. 2. The Dynkin diagram of $\tilde{\mathfrak {g}}$ can obtained from that of $\bar{\mathfrak {g}}$ by removing the r nodes in the $\sigma -$orbit ${\bar{I}}_0$ of the 0-th node. Due to the numbering of the Dynkin diagrams we have chosen, the set ${\tilde{I}}$ of nodes of the Dynkin diagram of $\tilde{\mathfrak {g}}$ can be obtained as

  $$\begin{aligned} {\tilde{I}}={\bar{I}}\setminus {\bar{I}}_0=\{1,\dots ,{\bar{n}}+1-r\}. \end{aligned}$$

• The Lie algebra $\mathfrak {g}=\tilde{\mathfrak {g}}^{(r)}$ is the affine Kac-Moody algebra introduced in Sect. 2. The Dynkin diagram of $\mathfrak {g}$ can be obtained by folding the Dynkin diagram of $\bar{\mathfrak {g}}$ under $\sigma$. If $r=1$ then $\sigma$ is the identity and $\mathfrak {g}=\bar{\mathfrak {g}}$, if $r>1$ then $\mathfrak {g}$ is a twisted affine Kac-Moody algebra. Denoting by $n+1$ the number of orbits, and due to the numbering of nodes in the Dynkin diagram, then the set I of nodes of the Dynkin diagram of $\mathfrak {g}$ can be obtained as a subset of ${\bar{I}}$:

  $$\begin{aligned} I=\{0,1,\dots ,n\}\subset {\bar{I}}. \end{aligned}$$

• The Lie algebra $\mathring{\mathfrak {g}}$ is simple Lie algebra introduced in Sect. 2. As already shown, its Dynkin diagram can be obtained by removing the 0th-vertex from the Dynkin diagram of $\mathfrak {g}$ or, equivalently, by folding the Dynkin diagram of $\tilde{\mathfrak {g}}$ by $\sigma$. The nodes of the Dynkin diagram of $\mathring{\mathfrak {g}}$ are indexed by the set

  $$\begin{aligned} \mathring{I}=I\setminus \{0\}=\{1,\dots ,n\}. \end{aligned}$$

Table 5 Dynkin diagram of $\bar{\mathfrak {g}}=\mathfrak {g}$ (case $r=1$)

Table 6 Dynkin diagram of $\bar{\mathfrak {g}}$ with the automorphism $\sigma$ (of order $r>1$), and the Dynkin diagram of the folded algebra $\mathfrak {g}$

1.1 The untwisted affine Kac-Moody algebra $\bar{\mathfrak {g}}$

Let $\bar{\mathfrak {g}}$ one of the Kac–Moody algebra of affine type in Table 4, with Dynkin diagram labeled as in Tables  and . Note that $\bar{\mathfrak {g}}$ is always an untwisted affine Lie algebra and it is simply-laced, namely its Cartan matrix is symmetric. Let ${\bar{I}}=\{0,\dots ,{\bar{n}}\}$ denote the set of nodes of the Dynkin diagram of $\bar{\mathfrak {g}}$. Let $\{{\bar{f}}_i,{\bar{\alpha }}^\vee _i,{\bar{e}}_i\,|\,i\in {\bar{I}}\}$ be Chevalley generators of $\bar{\mathfrak {g}}$, satisfying the relations ($i,j\in {\bar{I}}$):

$$\begin{aligned} {[}{\bar{\alpha }}^\vee _i,{\bar{\alpha }}^\vee _j]=0,\quad [{\bar{\alpha }}^\vee _i,{\bar{e}}_j]={\bar{C}}_{ij}{\bar{e}}_j,\quad [{\bar{\alpha }}^\vee _i,{\bar{f}}_j]=-{\bar{C}}_{ij}{\bar{f}}_j,\quad [{\bar{e}}_i,{\bar{f}}_j]=\delta _{ij}{\bar{\alpha }}^\vee _i, \end{aligned}$$

where ${\bar{C}}=({\bar{C}}_{ij})_{i,j\in {\bar{I}}}$ is the Cartan matrix of $\bar{\mathfrak {g}}$. We define

$$\begin{aligned} f=\sum _{i\in {\bar{I}}}{\bar{f}}_i\in \bar{\mathfrak {g}}. \end{aligned}$$

(A.1)

Fix a Cartan subalgebra $\bar{\mathfrak {h}}$ of $\bar{\mathfrak {g}}$ such that $\{{\bar{\alpha }}^\vee _i\,|\,i\in {\bar{I}}\}\subset \bar{\mathfrak {h}}$ is the set of simple coroots and denote by $\{{\bar{\alpha }}_i\,|\,i\in {\bar{I}}\}\subset \bar{\mathfrak {h}}^*$ the corresponding set of simple roots, such that $\langle {\bar{\alpha }}^\vee _i,{\bar{\alpha }}_j\rangle ={\bar{C}}_{ij}$. Fix a scaling element ${\bar{\textbf{d}}}\in \bar{\mathfrak {h}}$ satisfying $\langle {\bar{\textbf{d}}},{\bar{\alpha }}_i\rangle =\delta _{i0}$. Then

$$\begin{aligned} \bar{\mathfrak {h}}=\langle {\bar{\alpha }}^\vee _0,\dots ,{\bar{\alpha }}^\vee _{{\bar{n}}},{\bar{\textbf{d}}}\rangle . \end{aligned}$$

(A.2)

Let $\{{\bar{a}}_i\,|\,i\in {\bar{I}}\}$ be Kac’s labels of $\bar{\mathfrak {g}}$ (these are equal to the dual Kac’s labels, since $\bar{\mathfrak {g}}$ is simply laced), namely a set of positive integers defined by the relations

$$\begin{aligned} \sum _{j\in {\bar{I}}}{\bar{C}}_{ij}{\bar{a}}_j=0,\qquad i\in {\bar{I}}, \end{aligned}$$

(A.3)

and which we normalize by setting ${\bar{a}}_0=1$. The element of $\bar{\mathfrak {h}}$ given by

$$\begin{aligned} K=\sum _{i\in {\bar{I}}}{\bar{a}}_i{\bar{\alpha }}^\vee _i \end{aligned}$$

(A.4)

is the canonical central element of $\bar{\mathfrak {g}}$. Denote by $\bar{\mathfrak {n}}^+$ (resp. $\bar{\mathfrak {n}}^-$) the subalgebra of $\bar{\mathfrak {g}}$ generated by $\{{\bar{e}}_i\,|\,i\in {\bar{I}}\}$ (resp. $\{{\bar{f}}_i\,|\,i\in {\bar{I}}\}$), so that $\bar{\mathfrak {g}}$ admits the triangular decomposition

$$\begin{aligned} \bar{\mathfrak {g}}=\bar{\mathfrak {n}}^-\oplus \bar{\mathfrak {h}}\oplus \bar{\mathfrak {n}}^+. \end{aligned}$$

(A.5)

Denote by $\bar{\mathfrak {b}}_\pm =\bar{\mathfrak {h}}\oplus \bar{\mathfrak {n}}_\pm$ the corresponding Borel subalgebras. Let ${\bar{\Delta }}\subset \bar{\mathfrak {h}}^*$ be the set of roots of $\bar{\mathfrak {g}}$, so that

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

(A.6)

with $\bar{\mathfrak {g}}_\alpha$ the corresponding root space. Let $\{{\bar{\omega }}^\vee _i\,|\,i\in {\bar{I}}\}\subset \bar{\mathfrak {h}}$ be fundamental coweights of $\bar{\mathfrak {g}}$, satisfying $\langle {\bar{\omega }}^\vee _i,{\bar{\alpha }}_j\rangle =\delta _{ij}$ ($i,j\in {\bar{I}}$). In particular, we set ${\bar{\omega }}^\vee _0={\bar{\textbf{d}}}$. Consider the element

$$\begin{aligned} \rho ^\vee =\sum _{i\in {\bar{I}}}{\bar{\omega }}^\vee _i\in P^\vee ({\bar{\mathfrak {g}}}). \end{aligned}$$

(A.7)

The adjoint action of $\rho ^\vee$ defines the principal gradation on $\bar{\mathfrak {g}}$:

$$\begin{aligned} \bar{\mathfrak {g}}=\bigoplus _{j\in \mathbb {Z}}\bar{\mathfrak {g}}^j,\qquad \bar{\mathfrak {g}}^j=\{x\in \bar{\mathfrak {g}}\,|\,[\rho ^\vee ,x]=jx\}. \end{aligned}$$

(A.8)

In particular,

$$\begin{aligned} \bar{\mathfrak {g}}^{-1}=\bigoplus _{j\in {\bar{I}}}\mathbb {C}{\bar{f}}_j, \end{aligned}$$

(A.9)

so that f defined in (A.1) belongs to $\bar{\mathfrak {g}}^{-1}$. For $k\in \mathbb {Z}$ we denote

$$\begin{aligned} \left( \bar{\mathfrak {g}}\right) ^{\ge k}=\bigoplus _{j\ge k}\bar{\mathfrak {g}}^j, \end{aligned}$$

(A.10)

from which it follows that $\left( \bar{\mathfrak {g}}\right) ^{\ge 1}=\bar{\mathfrak {n}}^+$, while $\left( \bar{\mathfrak {g}}\right) ^{\ge 0}=\bar{\mathfrak {h}}\oplus \bar{\mathfrak {n}}^+=\bar{\mathfrak {b}}^+$ and

$$\begin{aligned} \left( \bar{\mathfrak {g}}\right) ^{\ge -1}=\left( \bigoplus _{j\in {\bar{I}}}\mathbb {C}{\bar{f}}_j\right) \oplus \bar{\mathfrak {b}}^+. \end{aligned}$$

(A.11)

1.2 Dynkin diagram automorphism of $\bar{\mathfrak {g}}$ and the folded affine algebra $\mathfrak {g}$

Let $\sigma$ be a Dynkin diagram automorphism of $\bar{\mathfrak {g}}$, namely a permutation of the set ${\bar{I}}$ such that ${\bar{C}}_{\sigma (i)\sigma (j)}={\bar{C}}_{ij}$. Extend $\sigma$ to a Lie algebra automorphism (still denoted by $\sigma$) $\sigma \in {{\,\textrm{Aut}\,}}(\bar{\mathfrak {g}})$ defined on Chevalley generators by ($i\in {\bar{I}}$)

$$\begin{aligned} \sigma ({\bar{e}}_i)={\bar{e}}_{\sigma (i)},\quad \sigma ({\bar{\alpha }}^\vee _i)={\bar{\alpha }}^\vee _{\sigma (i)},\quad \sigma ({\bar{f}}_i)={\bar{f}}_{\sigma (i)}. \end{aligned}$$

(A.12)

The automorphism $\sigma$ induces the following gradation on $\bar{\mathfrak {g}}$:

$$\begin{aligned} \bar{\mathfrak {g}}=\bigoplus _{\ell =0}^{r-1}\bar{\mathfrak {g}}_\ell ,\quad \bar{\mathfrak {g}}_\ell =\left\{ x\in \bar{\mathfrak {g}}\,|\,\sigma (x)=\varepsilon ^\ell x\right\} \end{aligned}$$

(A.13)

where

$$\begin{aligned} \varepsilon =e^\frac{2\pi i}{r} \end{aligned}$$

(A.14)

and r is the order of $\sigma$.

Remark A.1

Dynkin diagram automorphisms of affine Kac-Moody algebras, whose folded diagram is the Dynkin diagram of an affine Kac-Moody algebra were considered by Liu, Wu, Zhang and Zhou [42]. It was proved in the same paper that the fixed point subalgebra $\bar{\mathfrak {g}}_0$ does not need to be isomorphic to the affine algebra $\mathfrak {g}$ associated to the folded diagram. Indeed, one has $\bar{\mathfrak {g}}_0=\mathfrak {g}\oplus \mathcal {H}$, where $\mathcal {H}$ is either $\{0\}$ or a proper vector subspace of the principal Heisenberg subalgebra of $\bar{\mathfrak {g}}$. The subspace $\mathcal {H}$ plays no role in the present construction, since (as shown below) the $\bar{\mathfrak {g}}_0-$part of the opers we are interested in lies in $\mathfrak {g}$.

The folded affine Lie algebra $\mathfrak {g}$ obtained from the Dynkin diagram of $\bar{\mathfrak {g}}$ is an affine algebra of type $\tilde{\mathfrak {g}}^{(r)}$ and therefore it is twisted if $r>1$, see Table . The structure of $\mathfrak {g}$ can be obtained from that of $\bar{\mathfrak {g}}$ as follows. We denote by ${\bar{I}}_j\subset {\bar{I}}$ the $\sigma$-orbit of $j\in {\bar{I}}$, and let $n+1$ be the number of orbits. Due to the numbering of the Dynkin diagrams in Table  we have that the nodes $0,\dots ,n$ lie in different orbits, so we can represent the set of orbits by $I=\{0\dots ,n\}\subset {\bar{I}}$. For $i\in {\bar{I}}$ we denote by $\langle i\rangle \in \mathbb {Z}^+$ the cardinality of the $\sigma$-orbit ${\bar{I}}_i$. The Cartan matrix $C=(C_{ij})_{i,j\in I}$ of $\mathfrak {g}$ can be obtained summing over the columns of ${\bar{C}}$ along the orbits of $\sigma$:

$$\begin{aligned} C_{ij}=\sum _{\ell =1}^{\langle i\rangle }{\bar{C}}_{\sigma ^\ell (i)j},\qquad i,j\in I. \end{aligned}$$

The elements

$$\begin{aligned} e_i=\sum _{\ell =1}^{\langle i\rangle }{\bar{e}}_{\sigma ^\ell (i)},\quad \alpha ^\vee _i=\sum _{\ell =1}^{\langle i\rangle }{\bar{\alpha }}^\vee _{\sigma ^\ell (i)},\quad f_i=\sum _{\ell =1}^{\langle i\rangle }{\bar{f}}_{\sigma ^\ell (i)},\qquad i\in I. \end{aligned}$$

(A.15)

satisfy the relations ($i,j\in I$)

$$\begin{aligned} {[}\alpha ^\vee _i,\alpha ^\vee _j]=0,\quad [\alpha ^\vee _i,e_j]=C_{ij}e_j,\quad [\alpha ^\vee _i,f_j]=-C_{ij}f_j,\quad [e_i,f_j]=\delta _{ij}\alpha ^\vee _i, \end{aligned}$$

together with the Serre relations [42], and are therefore Chevalley generators of $\mathfrak {g}$. In addition, the elements

$$\begin{aligned} \alpha _i=\frac{1}{\langle i\rangle }\sum _{\ell =1}^{\langle i\rangle }{\bar{\alpha }}_{\sigma ^\ell (i)},\qquad i\in I \end{aligned}$$

(A.16)

are simple roots of $\mathfrak {g}$, namely they satisfy $\langle \alpha _i^\vee ,\alpha _j\rangle =C_{ij}$ ($i,j\in I$). The following elementary result will be useful later:

Lemma A.2

Recall the Kac’s labels $\{{\bar{a}}_i\,|\,i\in {\bar{I}}\}$ of $\bar{\mathfrak {g}}$. The quantities

$$\begin{aligned}a_i= \frac{\langle i\rangle }{r}{\bar{a}}_i,\qquad a^\vee _i= {\bar{a}}_i,\qquad i\in I\end{aligned}$$

are, respectively, Kac’s labels and dual Kac’s labels of $\mathfrak {g}$, satisfying the relations (2.1). In particular, $a_0=a^\vee _0=1$.

Proof

From the relation (A.3) and the invariance of ${\bar{C}}$ under $\sigma$ it follows that the ${\bar{a}}_i$’s are $\sigma -$invariant: ${\bar{a}}_{\sigma (i)}={\bar{a}}_i$ ($i\in {\bar{I}}$). Moreover, from (A.3) for $i\in {\bar{I}}$ one has

$$\begin{aligned} 0=\sum _{j\in {\bar{I}}}{\bar{C}}_{ij}{\bar{a}}_j=\sum _{j\in I}\sum _{\ell =1}^{\langle j \rangle }{\bar{C}}_{i,\sigma ^\ell (j)}{\bar{a}}_{\sigma ^\ell (j)}=\sum _{j\in I}\left( \sum _{\ell =1}^{\langle j \rangle }{\bar{C}}_{i,\sigma ^\ell (j)}\right) {\bar{a}}_{j}, \end{aligned}$$

and summing over the i-th orbit we get ($i\in I$):

$$\begin{aligned} 0&=\sum _{m= 1}^{\langle i\rangle }\sum _{j\in {\bar{I}}}{\bar{C}}_{\sigma ^m(i),j}{\bar{a}}_j=\sum _{m= 1}^{\langle i\rangle }\sum _{j\in I}\left( \sum _{\ell =1}^{\langle j \rangle }{\bar{C}}_{\sigma ^m(i),\sigma ^\ell (j)}\right) {\bar{a}}_{j}\\&=\sum _{j\in I}\sum _{\ell =1}^{\langle j \rangle }\left( \sum _{m= 1}^{\langle i\rangle }{\bar{C}}_{\sigma ^{m-\ell }(i),j}\right) {\bar{a}}_{j}=\sum _{j\in I} C_{ij}\langle j \rangle {\bar{a}}_{j}. \end{aligned}$$

By a direct (case-by-case) inspection we obtain that $\langle 0\rangle =r$, so that the quantities $a_j=\langle j \rangle {\bar{a}}_j/r$ ($j\in I$) satisfy (2.1) and $a_0=1$. For the $a^\vee _i$’s the proof is similar. $\square$

We set

$$\begin{aligned} D_i=\frac{a_i}{a^\vee _i},\qquad i\in I. \end{aligned}$$

(A.17)

Due to the above lemma, this is consistent with (2.8). We now show that certain elements of $\bar{\mathfrak {g}}$ introduced above are $\sigma$-invariant, and - moreover - they belong to $\mathfrak {g}\subset \bar{\mathfrak {g}}$.

• i) The element $f\in \bar{\mathfrak {g}}$ defined in (A.1) is $\sigma$-invariant (i.e. $\sigma (f)=f$), and recalling the elements $f_i\in \mathfrak {g}$ introduced in (A.15) we have

  $$\begin{aligned} f=\sum _{i\in {\bar{I}}}{\bar{f}}_i=\sum _{j\in I}\left( \sum _{\ell =1}^{\langle j\rangle }{\bar{f}}_{\sigma ^\ell (j)}\right) =\sum _{j\in I}f_j \end{aligned}$$

  (A.18)

  from which it follows that $f\in \mathfrak {g}$. In particular, it coincides with (2.23).

• ii) Similarly, the canonical central element $K\in \bar{\mathfrak {h}}$ of $\bar{\mathfrak {g}}$ given by (A.4) is $\sigma$-invariant ($\sigma (K)=K$) and, using Lemma A.2, it turns out to be equal to the canonical central element $K\in \mathfrak {h}$ of $\mathfrak {g}$:

  $$\begin{aligned} K=\sum _{i\in I}a^\vee _i \alpha ^\vee _i. \end{aligned}$$

  (A.19)

• iii) The fundamental coweights $\{\omega ^\vee _i\,|\,i\in I\}$ of $\mathfrak {g}$ can be written in terms of those of $\bar{\mathfrak {g}}$ as

  $$\begin{aligned} \omega ^\vee _i=\sum _{\ell =1}^{\langle i\rangle }{\bar{\omega }}^\vee _{\sigma ^\ell (i)},\qquad i\in I, \end{aligned}$$

  (A.20)

  and they satisfy $\langle \omega _i^\vee ,\alpha _j\rangle =\delta _{ij}$ ($i,j\in I$). We denote the coweight lattice and the coroot lattice of $\mathfrak {g}$, respectively, as

  $$\begin{aligned} P^\vee (\mathfrak {g})=\bigoplus _{i\in I}\mathbb {Z}\omega ^\vee _i,\qquad Q^\vee (\mathfrak {g})=\bigoplus _{i\in I}\mathbb {Z}\alpha ^\vee _i. \end{aligned}$$

  Recalling that $\langle 0\rangle =r$, we denote

  $$\begin{aligned} \textbf{d}=\omega ^\vee _0=\sum _{\ell =1}^{r}{\bar{\omega }}^\vee _{\sigma ^\ell (0)} ={\bar{\textbf{d}}}+\sum _{\ell =2}^{r}{\bar{\omega }}^\vee _{\sigma ^\ell (0)}, \end{aligned}$$

  (A.21)

  where ${\bar{\textbf{d}}}$ is the scaling element of $\bar{\mathfrak {g}}$. The element $\textbf{d}\in P^\vee (\mathfrak {g})$, is manifestly $\sigma -$invariant, satisfies $\langle \textbf{d},\alpha _j\rangle =\delta _{0j}$ ($j\in I$), and it is therefore a scaling element of $\mathfrak {g}$. Due to (A.21) we can write the Cartan subalgebra (A.2) of $\bar{\mathfrak {h}}$ equivalently as

  $$\begin{aligned} \bar{\mathfrak {h}}=\langle {\bar{\alpha }}^\vee _0,\dots ,{\bar{\alpha }}^\vee _{{\bar{n}}},\textbf{d}\rangle . \end{aligned}$$

  (A.22)

  The Cartan subalgebra $\mathfrak {h}=\bar{\mathfrak {h}}\cap \mathfrak {g}$ of $\mathfrak {g}$ is thus given by

  $$\begin{aligned} \mathfrak {h}=\langle \alpha ^\vee _0,\dots ,\alpha ^\vee _n,\textbf{d}\rangle , \end{aligned}$$

  (A.23)

  where the $\alpha ^\vee _i$’s are given by (A.15).

• iv) The element $\rho ^\vee \in \bar{\mathfrak {h}}$ defined by (A.7) is fixed by $\sigma$ (i.e. $\sigma (\rho ^\vee )=\rho ^\vee$) and it can be written as the sum of the fundamental coweights of $\mathfrak {g}$:

  $$\begin{aligned} \rho ^\vee =\sum _{i\in I}\omega ^\vee _i. \end{aligned}$$

  (A.24)

  In particular, $\rho ^\vee \in P^\vee (\mathfrak {g})\subset \mathfrak {h}$.

1.3 The simple Lie algebras $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$ as subalgebras of $\bar{\mathfrak {g}}$

We can recover the simple Lie algebras $\tilde{\mathfrak {g}}$ and $\mathring{\mathfrak {g}}$ introduced in Sect. 2 as subalgebras of $\bar{\mathfrak {g}}$ and $\mathfrak {g}$. First, note that we can write the set ${\tilde{I}}$, the nodes of $\tilde{\mathfrak {g}}$ as ${\tilde{I}}={\bar{I}}\setminus {\bar{I}}_0$. In other words, ${\tilde{I}}$ is the subset of nodes in the Dynkin diagram of $\bar{\mathfrak {g}}$ with the orbit of the $0-$th node removed. The subalgebra of $\bar{\mathfrak {g}}$ generated by $\{{\bar{e}}_i,{\bar{\alpha }}^\vee _i,{\bar{f}}_i\,|\,i\in {\tilde{I}}\}$ is isomorphic to the simple finite dimensional Lie algebra $\tilde{\mathfrak {g}}$; in particular, ${\bar{C}}_{ij}={\tilde{C}}_{ij}$ ($i,j\in {\tilde{I}}$). We fix this isomorphims by identifying ${\tilde{e}}_i={\bar{e}}_i$, $\tilde{\alpha _i}^\vee ={\bar{\alpha }}_i^\vee$, ${\tilde{f}}_i={\bar{f}}_i$ ($i\in {\tilde{I}}$). Since we obtained $\mathring{\mathfrak {g}}$ as a subalgebra of $\tilde{\mathfrak {g}}$ we obtain the following inclusions of Lie subalgebras:

$$\begin{aligned} \mathring{\mathfrak {g}}\subseteq \tilde{\mathfrak {g}}\subset \bar{\mathfrak {g}}, \end{aligned}$$

where $\mathring{\mathfrak {g}}= \tilde{\mathfrak {g}}$ in the $r=1$ case. On the other hand, the Dynkin diagram of the simple Lie algebra $\mathring{\mathfrak {g}}$ can also be obtained from the Dynkin diagram of the affine algebra $\mathfrak {g}$ by removing the $0-$th vertex ($\mathring{I}=I\setminus \{0\}$) so that we can view $\mathring{\mathfrak {g}}$ as a subalgebra of $\mathfrak {g}$. Since $\mathfrak {g}$ is a subalgebra of $\tilde{\mathfrak {g}}$, we obtain another set of inclusions

$$\begin{aligned} \mathring{\mathfrak {g}}\subset \mathfrak {g}\subseteq \bar{\mathfrak {g}}, \end{aligned}$$

where $\mathfrak {g}=\bar{\mathfrak {g}}$ in the $r=1$ case. The above inclusions are consistent with the action of $\sigma$ and with the triangular decomposition: the restriction of the Dynkin diagram automorphism $\sigma \in {{\,\textrm{Aut}\,}}(\bar{\mathfrak {g}})$ to the subalgebra $\mathring{\mathfrak {g}}$ coincides with the Dynkin diagram automorphism $\sigma \in {{\,\textrm{Aut}\,}}(\tilde{\mathfrak {g}})$ introduced in Sect. 2, and we have the inclusions

$$\begin{aligned}{} & {} \mathring{\mathfrak {n}}^\pm \subset \tilde{\mathfrak {n}}^\pm \subset \bar{\mathfrak {n}}^\pm ,\qquad \mathring{\mathfrak {h}}\subset \tilde{\mathfrak {h}}\subset \bar{\mathfrak {h}}, \end{aligned}$$

(A.25)

$$\begin{aligned}{} & {} \mathring{\mathfrak {n}}^\pm \subset \mathfrak {n}^\pm \subset \bar{\mathfrak {n}}^\pm ,\qquad \mathring{\mathfrak {h}}\subset \mathfrak {h}\subset \bar{\mathfrak {h}}. \end{aligned}$$

(A.26)

We finally recall that if $\{\mathring{\omega }^\vee _i | i\in \mathring{I}\}$ be fundamental coweights of $\mathring{\mathfrak {g}}$, defined by the relations $\langle \mathring{\omega }^\vee _j,\alpha _j\rangle =\delta _{ij}$ ($i,j\in \mathring{I}$), then for $i\in \mathring{I}$ one has $\omega ^\vee _i=\mathring{\omega }^\vee _i+a_i\textbf{d}$ where the $\omega ^\vee _i$s are fundamental coweights of $\mathfrak {g}$ given in (A.20) and $\textbf{d}$ is the scaling element (A.21) of $\mathfrak {g}$. The relation between the elements (2.15) and (A.24) is therefore given by

$$\begin{aligned} \rho ^\vee =\mathring{\rho }^\vee +h\textbf{d}, \end{aligned}$$

(A.27)

where h is the Coxeter number of $\mathfrak {g}$.

1.4 A parabolic subalgebra of $\bar{\mathfrak {g}}$

Let $\tilde{\mathfrak {n}}^+$ be the nilpotent Lie algebra defined by (2.4). Due to (A.25) we can view $\tilde{\mathfrak {n}}^+$ as a subalgebra of $\bar{\mathfrak {n}}^+$, and more precisely as the subalgebra generated by the elements $\{{\tilde{e}}_i={\bar{e}}_i|i\in {\tilde{I}}\}$. Given the triangular decomposition (A.5) of $\bar{\mathfrak {g}}$, and recalling that $\bar{\mathfrak {b}}^-=\bar{\mathfrak {n}}^-\oplus \bar{\mathfrak {h}}$, we define the following subalgebra of $\bar{\mathfrak {g}}$:

$$\begin{aligned} \bar{\mathfrak {p}}=\bar{\mathfrak {b}}^-\oplus \tilde{\mathfrak {n}}^+. \end{aligned}$$

(A.28)

This is a parabolic subalgebra of $\bar{\mathfrak {g}}$, standard with respect to the Borel subalgebra $\bar{\mathfrak {b}}^-$, and it is stable under the action of $\sigma$:

$$\begin{aligned} x\in \bar{\mathfrak {p}}\quad \Longleftrightarrow \quad \sigma x\in \bar{\mathfrak {p}}. \end{aligned}$$

In view of the application to the theory of opers, we are interested only in the intersection of $\bar{\mathfrak {p}}$ with the subspace (A.11) of $\bar{\mathfrak {g}}$.

Lemma A.3

• i) We have

  $$\begin{aligned} \bar{\mathfrak {p}}\cap \left( \bar{\mathfrak {g}}\right) ^{\ge -1}=\bar{\mathfrak {g}}^{-1}\oplus \bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+, \end{aligned}$$

  (A.29)

  where $\bar{\mathfrak {g}}^{-1}$ is given by (A.9) and $\bar{\mathfrak {h}}$ is the Cartan subalgebra (A.22) of $\bar{\mathfrak {g}}$. In particular, $\bar{\mathfrak {p}}\cap \left( \bar{\mathfrak {g}}\right) ^{\ge -1}$ is a finite dimensional $\mathbb {C}-$vector space.

• ii) The Cartan subalgebra $\bar{\mathfrak {h}}$ of $\bar{\mathfrak {g}}$ decomposes as

  $$\begin{aligned} \bar{\mathfrak {h}}=\mathbb {C}\textbf{d}\oplus \mathbb {C}\tilde{\mathfrak {h}}\oplus \bar{\mathfrak {h}}', \end{aligned}$$

  (A.30)

  where

  $$\begin{aligned} \bar{\mathfrak {h}}'=\langle {\bar{\alpha }}^\vee _{\sigma ^\ell (0)}\,|\,\ell =1,\dots ,r\rangle , \end{aligned}$$

  (A.31)

  and $\tilde{\mathfrak {h}}$ is the Cartan subalgebra (2.2) of $\tilde{\mathfrak {g}}$.

Proof

• i) Obvious from (A.11) and (A.9), and the fact that $\tilde{\mathfrak {n}}^+\subset \bar{\mathfrak {b}}^+$.

• ii) The Cartan subalgebra $\bar{\mathfrak {h}}$ has a basis given by (A.22). By definition, the coroots in the $\sigma$-orbit of 0 are a basis of $\bar{\mathfrak {h}}'$, while from (2.2) the remaining coroots form a basis of $\tilde{\mathfrak {h}}$.

$\square$

Proposition A.4

The subalgebra $\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+$ of $\bar{\mathfrak {g}}$ decomposes with respect to the gradation (A.13) as

$$\begin{aligned} \bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+=\bigoplus _{\ell =0}^{r-1}\mathfrak {m}_\ell , \end{aligned}$$

(A.32)

where $\mathfrak {m}_\ell =(\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+)\cap \bar{\mathfrak {g}}_\ell$ is given by

$$\begin{aligned} \mathfrak {m}_\ell = {\left\{ \begin{array}{ll} \mathfrak {h}\oplus \mathring{\mathfrak {n}}^+,&{} \ell =0\\ \bar{\mathfrak {h}}'_\ell \oplus \tilde{\mathfrak {b}}^+_\ell ,&{} \ell \ge 1. \end{array}\right. } \end{aligned}$$

In the formula above, $\mathfrak {h}$ is the Cartan subalgebra (A.23) of $\mathfrak {g}$ and $\mathring{\mathfrak {n}}^+$ is given by (2.11). Moreover, $\bar{\mathfrak {h}}'_\ell =\bar{\mathfrak {h}}'\cap \bar{\mathfrak {g}}_\ell$, where $\bar{\mathfrak {h}}'$ is given by (A.31), and $\tilde{\mathfrak {b}}^+_\ell =\tilde{\mathfrak {b}}^+\cap \tilde{\mathfrak {g}}_\ell$.

Proof

Since $\bar{\mathfrak {h}}$ decomposes as in (A.30), we can write $\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+=\mathbb {C}\textbf{d}\oplus \bar{\mathfrak {h}}'\oplus \tilde{\mathfrak {b}}^+$. The Borel subalgebra $\tilde{\mathfrak {b}}^+$ can be written as

$$\begin{aligned} \tilde{\mathfrak {b}}^+=\bigoplus _{\ell =0}^{r-1}\tilde{\mathfrak {b}}^+_\ell \end{aligned}$$

where in particular $\tilde{\mathfrak {b}}^+_0=\mathring{\mathfrak {b}}^+$. On the other hand, the subspace $\bar{\mathfrak {h}}'$ given by (A.31) decomposes as

$$\begin{aligned} \bar{\mathfrak {h}}'=\bigoplus _{\ell =0}^{r-1}\bar{\mathfrak {h}}'_\ell , \end{aligned}$$

where each space $\bar{\mathfrak {h}}'_\ell$ is 1-dimensional, generated by

$$\begin{aligned} \kappa _\ell =\sum _{m=0}^{r-1}\varepsilon ^{-m\ell }{\bar{\alpha }}^\vee _{\sigma ^m(0)},\qquad \ell =0,\dots ,r-1. \end{aligned}$$

In particular, $\kappa _0=\alpha ^\vee _0.$ The scaling element $\textbf{d}$ satisfies $\sigma (\textbf{d})=\textbf{d}$, so that $\mathbb {C}\textbf{d}\in \mathfrak {m}_0$. Thus for $\ell \ge 1$ we get $\mathfrak {m}_\ell =(\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+)\cap \bar{\mathfrak {g}}_\ell =\bar{\mathfrak {h}}'_\ell \oplus \tilde{\mathfrak {b}}^+_\ell$. On the other hand, $\mathfrak {m}_0=(\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+)\cap \bar{\mathfrak {g}}_0=(\mathbb {C}\textbf{d}\oplus \bar{\mathfrak {h}}' \oplus \tilde{\mathfrak {b}}^+)\cap \bar{\mathfrak {g}}_0=\mathbb {C}\textbf{d}\oplus \mathbb {C}\alpha ^\vee _0\oplus (\tilde{\mathfrak {b}}^+\cap \bar{\mathfrak {g}}_0)$. Since $\tilde{\mathfrak {b}}^+\subset \tilde{\mathfrak {g}}$ and $\bar{\mathfrak {g}}_0\cap \tilde{\mathfrak {g}}=\tilde{\mathfrak {g}}_0$, then $\tilde{\mathfrak {b}}^+\cap \bar{\mathfrak {g}}_0=\tilde{\mathfrak {b}}^+\cap \tilde{\mathfrak {g}}_0=\mathring{\mathfrak {b}}^+=\mathring{\mathfrak {h}}\oplus \mathring{\mathfrak {n}}^+$, where we used (2.11). So $\mathfrak {m}_0=\mathbb {C}\textbf{d}\oplus \mathbb {C}\alpha ^\vee _0\oplus \mathring{\mathfrak {h}}\oplus \mathring{\mathfrak {n}}^+ =\langle \textbf{d},\alpha ^\vee _0,\alpha ^\vee _1,\dots ,\alpha ^\vee _n\rangle \oplus \mathring{\mathfrak {n}}^+=\mathfrak {h}\oplus \mathring{\mathfrak {n}}^+$. $\square$

The following results are an immediate consequence of the above proposition.

Corollary A.5

Let $f\in \mathfrak {g}\subset \bar{\mathfrak {g}}$ be given by (A.18).

• i) The affine subspace $f+\mathfrak {m}_0=f+\mathfrak {h}\oplus \mathring{\mathfrak {n}}^+$ of $\bar{\mathfrak {g}}$ is contained in $\mathfrak {g}$. In particular, each element of $f+\mathfrak {m}_0$ is fixed by $\sigma$. Moreover, $f+\mathfrak {m}_0$ is stable under the adjoint action of $\mathring{\mathfrak {n}}^+$.

• ii) For each $\ell \ge 1$ the subspace $\mathfrak {m}_\ell =\bar{\mathfrak {h}}'_\ell \oplus \tilde{\mathfrak {b}}^+_\ell$ is a $\mathfrak {m}_0$-module with the adjoint action.

• iii) The vector space $\bar{\mathfrak {g}}^{-1}\oplus \bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+$ is a $\mathfrak {m}_0$-module with the adjoint action.

• iv) The affine space $f+\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+$ is stable under the adjoint action of $\mathring{\mathfrak {n}}^+$.

1.5 Twisted parabolic Miura $\mathfrak {g}$-opers over $\mathbb {C}^*$

We consider twisted parabolic Miura $\mathfrak {g}$-opers for the class of affine algebras $\bar{\mathfrak {g}}$ listed in Table 4 and choosing $\bar{\mathfrak {p}}\subset \bar{\mathfrak {g}}$ given in (A.28) as parabolic subalgebra. The construction can be generalized in the obvious way to other Lie algebras and different choices of parabolic subalgebras.

Remark A.6

The notion of affine twisted parabolic Miura opers builds upon several other notions of classes of opers, which we briefly recall. The original notion of opers (with values in a finite dimensional simple Lie algebra) appeared for the first time in Drinfeld and Sokolov [22] and was later formalized by Beilinson and Drinfeld [10]. Affine opers were introduced by Ben-Zvi and Frenkel [11] (see also [25]) as suitable analogues of opers with values in an (affine) Kac–Moody algebra. Parabolic Miura opers were first considered in [25] as generalization to the parabolic case of the notion of Miura opers [11]. Finally, twisted opers were defined—in the finite dimensional case—by Frenkel and Gross [28].

Recall from Sect. 6 that we denoted by $\mathcal {K}$ the field of meromorphic functions on $\mathbb {C}^*$, and by $V(\mathcal {K})$ the space of meromorphic functions from $\mathbb {C}^*$ to a vector space V. Recall moreover the principal gradation (A.8) of $\bar{\mathfrak {g}}$, the associated filtration (A.10), and the element $f\in \bar{\mathfrak {g}}^{-1}$ given by (A.1) or equivalently by (A.18). Let moreover $\bar{\mathfrak {p}}$ be the parabolic subalgebra of $\bar{\mathfrak {g}}$ defined in (A.28). We introduce a set of operators taking values in $(\bar{\mathfrak {g}})^{\ge -1}\cap \bar{\mathfrak {p}}$ or equivalently, due to (A.29), with values in $\bar{\mathfrak {g}}^{-1}\oplus \bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+$. By Lemma A.3, this is a finite dimensional vector space.

Definition A.7

We denote by ${{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ the set of operators

$$\begin{aligned} L(z)=\partial _z+\frac{1}{z}\left( f+b(z)\right) ,\qquad b\in (\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+)(\mathcal {K}). \end{aligned}$$

(A.33)

Following [28], we define twisted opers by considering a class of operators in ${{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ which are fixed under a transformation which involves both the automorphism $\sigma$ and a rotation in $\mathbb {C}^*$. We first consider the two actions separately:

Lemma A.8

The set ${{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ is stable under the automorphism of $\mathbb {C}^*$ given by $z\mapsto az$ ($a\in \mathbb {C}^*)$. Indeed, this map induces the transformation

$$\begin{aligned} \partial _z+\frac{1}{z}\left( f+b(z)\right) \longrightarrow \partial _z+\frac{1}{z}\left( f+b(az)\right) . \end{aligned}$$

(A.34)

in ${{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$.

Proof

Under a local change of coordinates $z=\varphi (x)$, the operator (A.33) becomes

$$\begin{aligned} (\varphi ^*L)(x)=\partial _x+\frac{\varphi '(x)}{\varphi (x)}(f+b(\varphi (x))). \end{aligned}$$

Choosing $\varphi (x)=ax$ ($a\in \mathbb {C}^*$) we get $(\varphi ^*L)(x)=\partial _x+\frac{1}{x}\left( f+b(ax)\right) \in {{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$. $\square$

We are interested in the special case $a=e^{2\pi it}$, with $t\in \mathbb {R}$.

Definition A.9

Let $L(z)\in {{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ be given by (A.33). For every $t\in \mathbb {R}$ we define the rotated operator $L_t(z)\in {{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ as

$$\begin{aligned} L_t(z)=\partial _z+\frac{1}{z}\left( f+b(e^{2\pi it}z)\right) . \end{aligned}$$

(A.35)

The requirement that $L(z)\in {{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ is meromorphic at $z=0$ is thus expressed by the condition

$$\begin{aligned} L_1(z)=L(z). \end{aligned}$$

(A.36)

We now consider the action of $\sigma$.

Definition A.10

Given an operator $L(z)=\partial _z+A(z)$ with $A\in \bar{\mathfrak {g}}(\mathcal {K})$ we define the twisted operator as $L^\sigma (z)=\partial _z+\sigma (A)(z)$. In particular, for $L(z)\in {{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ as in (A.33), and if $b=\sum _{\ell =0}^{r-1}b_\ell$ according to the decomposition (A.32), with $b_\ell \in \mathfrak {m}_\ell$, then

$$\begin{aligned} L^\sigma (z)=\partial _z+\frac{1}{z}\left( f+\sum _{\ell =0}^{r-1}\varepsilon ^\ell b_\ell (z)\right) , \end{aligned}$$

(A.37)

where $\varepsilon$ is given by (A.14).

We extend the action of $\sigma$ from $\bar{\mathfrak {g}}$ to $\mathbb {C}^*\times \bar{\mathfrak {g}}$ by considering the map

$$\begin{aligned} (z,x)\mapsto (\varepsilon ^{-1}z,\sigma (x)),\qquad z\in \mathbb {C}^*,x\in \bar{\mathfrak {g}}, \end{aligned}$$

which induces the transformation

$$\begin{aligned} L(z)\mapsto L^\sigma _{-\frac{1}{r}}(z) \end{aligned}$$

on ${{\,\textrm{op}\,}}_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$, and we restrict to the class of operators (A.33) which are pointwise fixed:

$$\begin{aligned} L^\sigma _{-\frac{1}{r}}(z)=L(z). \end{aligned}$$

(A.38)

Using Proposition A.4, we obtain that these invariant operators are of the form

$$\begin{aligned} L(z)=\partial _z+\frac{1}{z}\left( f+\sum _{\ell =0}^{r-1}z^\ell b_\ell (z^r)\right) , \end{aligned}$$

(A.39)

where

$$\begin{aligned} b_0&\in \mathfrak {m}_0(\mathcal {K})=\mathfrak {h}(\mathcal {K})\oplus \mathring{\mathfrak {n}}^+(\mathcal {K})\\ b_\ell&\in \mathfrak {m}_\ell (\mathcal {K})= \bar{\mathfrak {h}}'_\ell (\mathcal {K})\oplus \tilde{\mathfrak {b}}^+_\ell (\mathcal {K}),\quad \ell \ge 1. \end{aligned}$$

Definition A.11

We denote by ${{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ the set of operators (A.39).

From Corollary A.5, the affine subspace $f+\bar{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}^+$ is stable under the adjoint action of $\mathring{\mathfrak {n}}^+$.

Definition A.12

Let $\mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$ be the Gauge group defined in (6.2). For every $y\in \mathring{\mathfrak {n}}^+(\mathcal {K})$ we define the action of $\exp ({{\,\textrm{ad}\,}}y)\in \mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$ on $L(z)=\partial _z+(f+b(z))/z\in {{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ as

$$\begin{aligned} \exp ({{\,\textrm{ad}\,}}y(z^r))L(z)=\partial _z+\frac{1}{z}\left( \sum _{k\ge 0}\frac{({{\,\textrm{ad}\,}}y)^k}{k!}\left( f+b(z)\right) -rz^{r}\sum _{k\ge 0}\frac{({{\,\textrm{ad}\,}}y)^k}{(k+1)!}\frac{dy}{dz}\right) . \end{aligned}$$

(A.40)

We now define meromorphic twisted parabolic Miura $\mathfrak {g}-$opers.

Definition A.13

The space of meromorphic twisted $(\bar{\mathfrak {g}},\bar{\mathfrak {p}})-$opers (or twisted parabolic Miura $\mathfrak {g}-opers$) on $\mathbb {C}^*$ is defined as

$$\begin{aligned}{{\,\textrm{Op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K}) ={{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})/\mathring{\mathcal {N}}^{\sigma }(\mathcal {K}).\end{aligned}$$

We denote by [L(z)] the oper corresponding to the operator L(z).

It is often useful [10, 40] to enlarge the class of operators ${{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ by allowing the action of a more general Gauge group. To this aim, provide the following

Definition A.14

Let $\mathcal {H}^{\sigma }(\mathcal {K})$ the abelian group generated by $\varphi (z^r)^\mu$ with $\varphi \in \mathcal {K}{\setminus }\{0\}$ and $\mu \in P^\vee ({\mathfrak {g}})$, the coweight lattice of $\mathfrak {g}$. Let $L(z)\in {{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ be given by $L(z)=\partial _z+(f+b(z))/z$, and decompose b as $b=b_0+\sum _{\alpha \in {\bar{\Delta }}}b_\alpha \in \bar{\mathfrak {g}}$ with $b_0\in \bar{\mathfrak {h}}(\mathcal {K})$ and $b_\alpha \in \bar{\mathfrak {g}}_\alpha (\mathcal {K})$. We define

$$\begin{aligned} \varphi (z^r)^{{{\,\textrm{ad}\,}}\mu } L(z)=\partial _z+\frac{1}{z}\left( \sum _{i\in I}\varphi (z^r)^{-\langle \mu ,\alpha _i\rangle }f_i-\frac{rz^{r}\varphi '(z^r)}{\varphi (z^r)}\mu +\sum _{\alpha \in {\bar{\Delta }}}\varphi (z^r)^{\langle \mu ,\alpha \rangle }b_\alpha (z)\right) . \end{aligned}$$

(A.41)

Note that the only element $\mathcal {H}^{\sigma }(\mathcal {K})$ leaving ${{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ invariant is the identity.

Definition A.15

We set ${\widetilde{{{\,\textrm{op}\,}}}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})=\mathcal {H}^{\sigma }(\mathcal {K}){{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$. In other words, each element in ${\widetilde{{{\,\textrm{op}\,}}}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ is of the form (A.41) for some $\varphi (z^r)^\mu$ in $\mathcal {H}^{\sigma }(\mathcal {K})$ and $L(z)\in {{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$.

Note that each element in ${\widetilde{{{\,\textrm{op}\,}}}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ takes values in $(\bar{\mathfrak {g}})^{\ge -1}\cap \bar{\mathfrak {p}}$. The action of $\mathcal {H}^{\sigma }(\mathcal {K})$ on $\bar{\mathfrak {g}}(\mathcal {K})$ induces an action on $\mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$, given by

$$\begin{aligned} \varphi ^{{{\,\textrm{ad}\,}}\mu }\exp (y)=\exp (\varphi ^{{{\,\textrm{ad}\,}}\mu } y),\quad y\in \mathring{\mathfrak {n}}^+(\mathcal {K}),\varphi \in \mathcal {K}\setminus \{0\},\mu \in P^\vee (\mathfrak {g}). \end{aligned}$$

Definition A.16

We denote by $\mathcal {B}^{\sigma }(\mathcal {K})=\mathcal {H}^{\sigma }(\mathcal {K})\rtimes \mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$ the Gauge group obtained as a semidirect product of $\mathcal {H}^{\sigma }(\mathcal {K})$ and $\mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$, with multiplication given by

$$\begin{aligned} (\exp (y_1),\phi _1^{\mu _1})(\exp (y_2),\phi _2^{\mu _2})=(\exp (y_1)\exp (\phi _1^{{{\,\textrm{ad}\,}}\mu _1}y_2),\phi _1^{\mu _1}\phi _2^{\mu _2}). \end{aligned}$$

Note that $\mathcal {H}^\sigma (\mathcal {K})$ is normal in $\mathcal {B}^\sigma (\mathcal {K})$. Moreover, for each $L(z)\in {\widetilde{{{\,\textrm{op}\,}}}}^\sigma _{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ there is a unique element in $\mathcal {H}^\sigma (\mathcal {K})$ mapping L(z) to ${{\,\textrm{op}\,}}^\sigma _{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$. It thus follows that there is a bijection between the sets ${{\,\textrm{op}\,}}^\sigma _{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})/\mathring{\mathcal {N}}^\sigma (\mathcal {K})$ and ${\widetilde{{{\,\textrm{op}\,}}}}^\sigma _{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})/\mathcal {B}^\sigma (\mathcal {K})$.

Remark A.17

Recall the element $\rho ^\vee \in P^\vee (\mathfrak {g})$ given by (A.27). Applying the Gauge $z^{-\rho ^\vee }\in \mathcal {H}^\sigma (\mathcal {K})$ to an operator of type (A.33) one gets an operator of the form $\partial _z+f+b(z)$, for some $b\in \mathfrak {h}(\mathcal {K})\oplus \mathring{\mathfrak {n}}^+(\mathcal {K})$. This Gauge is often useful when dealing with opers [10]; we prefer however to use the Gauge (A.33) since it is invariant under the map $z\mapsto az$, see Lemma A.8. The class of operators of the form (A.33) looks similar to the notion of $(\le 1)$-singular opers, introduced by Beilinson and Drinfeld [10, Sec. 3.8.8]. Note, however, that in the present construction we allow the functions appearing in operators as well as the elements of the Gauge group to be meromorphic (over $\mathbb {C}^*$).

1.6 FFH $\mathfrak {g}$-opers are affine twisted parabolic Miura $\mathfrak {g}$-opers

We now prove that the opers ${{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$, introduced in Definition 6.1, are a particular class of affine twisted parabolic Miura opers. Since by Definition 6.5 the FFH $\mathfrak {g}$-opers belong to ${{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$, this implies that FFH $\mathfrak {g}$-opers are affine twisted parabolic Miura $\mathfrak {g}$-opers.

Proposition A.18

${{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})\subset {{\,\textrm{Op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$.

Proof

We first prove the inclusion ${{\,\textrm{op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})\subset {{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$. The operators in ${{\,\textrm{op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$ are of the form (6.1), and therefore they take values in $f\oplus \mathbb {C}\textbf{d}\oplus \mathring{\mathfrak {b}}^+\oplus \mathbb {C}v_\theta$. On the other hand, the operators in ${{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ are of the form (A.39), and since $\mathbb {C}\textbf{d}\oplus \mathring{\mathfrak {b}}^+\oplus \mathbb {C}K=\mathfrak {h}\oplus \mathring{\mathfrak {n}}^+$ and $v_\theta \in \tilde{\mathfrak {b}}^+_1$ we obtained the desired inclusion. The action on ${{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ of the Gauge group $\mathring{\mathcal {N}}^\sigma (\mathcal {K})$ given in Definition A.12 reduces to (6.3), (6.4) when restricted to ${{\,\textrm{op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})$. It then follows that ${{\,\textrm{Op}\,}}^{\sigma }_{\mathfrak {g},v_\theta }(\mathcal {K})\subset {{\,\textrm{Op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$. $\square$

Using the construction of affine twisted parabolic opers described in this Appendix, we can represents the operators (6.1) (as well as the corresponding opers) in more concrete terms by means of the loop realization (2.20) of the Kac-Moody algebra $\mathfrak {g}$. Let therefore $L(z)\in {{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},v_\theta }(\mathcal {K})\subset {{\,\textrm{op}\,}}^{\sigma }_{\bar{\mathfrak {g}},\bar{\mathfrak {p}}}(\mathcal {K})$ be given by (6.1). Let ${\tilde{\xi }}(z)$ satisfy ${\tilde{\xi }}'(z)=\frac{\xi (z)}{rz}$, recall that $\textbf{d}\in P^\vee (\mathfrak {g})$, and define

$$\begin{aligned} L(z;\lambda )=e^{{\tilde{\xi }}(z^r){{\,\textrm{ad}\,}}\textbf{d}}L(z), \end{aligned}$$

(A.42)

where we used the Gauge (A.41). Explicitly, we have:

$$\begin{aligned} L(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f}+b(z^r)+\left( zc(z^r)+e^{-{\tilde{\xi }}(z^r)}\lambda \right) v_\theta \right) , \end{aligned}$$

(A.43)

we call the above the loop realization of the operator L(z). The Gauge group $\mathring{\mathcal {N}}^{\sigma }(\mathcal {K})$ acts on the loop realization (A.43), so that we can define the oper $[L(z;\lambda )]$ corresponding to (A.43) as its equivalence class. Moreover, since $[\textbf{d},\mathring{\mathfrak {n}}^+]=0$ we have

$$\begin{aligned} {[}L(z;\lambda )]=[e^{{\tilde{\xi }}(z^r){{\,\textrm{ad}\,}}\textbf{d}}L(z)]=e^{{\tilde{\xi }}(z^r){{\,\textrm{ad}\,}}\textbf{d}}[L(z)], \end{aligned}$$

which provides a bijection between the equivalence classes, namely between the opers. From Proposition 6.4 the following result follows.

Corollary A.19

Let $\mathfrak {s}\subset \mathring{\mathfrak {b}}^+$ be a transversal subspace, as in (6.5). For each $L(z;\lambda )$ as in (A.43) the oper $[L(z,\lambda )]$ admits a unique canonical form

$$\begin{aligned} L_{\mathfrak {s}}(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f}+s(z^r)+\left( zc(z^r)+e^{-{\tilde{\xi }}(z^r)}\lambda \right) v_\theta \right) , \end{aligned}$$

for some $s\in \mathfrak {s}(\mathcal {K})$.

Note that the term $e^{-{\tilde{\xi }}(z^r)}$ in (A.43) does not need to be meromorphic. In particular, under the map $z\mapsto e^{2\pi i}z$ we have

$$\begin{aligned} e^{-{\tilde{\xi }}(z^r)}\mapsto e^{-2\pi i\xi (0)}e^{-{\tilde{\xi }}(z^r)}. \end{aligned}$$

(A.44)

In order to deal with this, for an operator $L(z;\lambda )$ of the form (A.43) and $t\in \mathbb {R}$ we define the rotated operator as the operator induced by the map

$$\begin{aligned} (z,\lambda )\mapsto (e^{2\pi it}z,e^{2\pi i\xi (0)t}\lambda ), \end{aligned}$$

that is, we set:

$$\begin{aligned} L_t(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f} +b(e^{2\pi irt}z^r)+\left( e^{2\pi it}zc(e^{2\pi irt}z^r)+e^{-{\tilde{\xi }}(e^{2\pi irt}z^r)}e^{2\pi i\xi (0)t}\lambda \right) v_\theta \right) . \end{aligned}$$

(A.45)

With this choice, the following property holds: the operator (6.1) is meromorphic at $z=0$ if and only if its loop realization (A.43) satisfies

$$\begin{aligned} L_1(z;\lambda )=L(z;\lambda ). \end{aligned}$$

(A.46)

We now consider in more detail the case when $\xi (x^r)$ is a nonzero constant, say $\xi (x^r)=-k$. This is the case of interest for the ODE/IM correspondence, see (3.7). The operator (6.1) becomes

$$\begin{aligned} L(z)=\partial _z+\frac{1}{z}\left( f-k\textbf{d}+b(z^r)+zc(z^r)v_\theta \right) , \end{aligned}$$

and it is meromorphic over $\mathbb {C}$, but its loop realization

$$\begin{aligned} L(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f}+b(z^r)+\left( zc(z^r)+z^{k}\lambda \right) v_\theta \right) , \end{aligned}$$

is meromorphic in $\mathbb {C}^*$ only. In this case, the rotated operator (A.45) takes the form

$$\begin{aligned} L_t(z;\lambda )=\partial _z+\frac{1}{z}\left( \mathring{f} +b(e^{2\pi irt}z^r)+\left( e^{2\pi it}zc(e^{2\pi irt}z^r)+z^{k}\lambda \right) v_\theta \right) , \end{aligned}$$

and it obviously satisfies the identity (A.46), which in the ODE/IM correspondence setting is known as Dorey-Tateo symmetry (cf. (3.9)).