An explicit construction of automorphic representations of the symplectic group with a given quadratic unipotent Arthur parameter

Abstract

To a large class of unipotent quadratic Arthur parameters for symplectic groups, we attach (explicitly realized in the space of square-integrable automorphic forms) an irreducible automorphic representation using degenerate Eisenstein series.

1 Introduction and preliminaries

1.1 Introduction

In this paper we explicitly realize, using degenerate Eisenstein series, an automorphic representation of a (global) symplectic group with a prescribed quadratic unipotent Arthur parameter. We manage to obtain this result for a large class of quadratic unipotent Arthur parameters, namely for those ones satisfying a technical condition $(\Delta )$ which is easy to check. In [1] is given a detailed study of parametrization of automorphic (square-integrable) representations of global symplectic group by these parameters, but without separation into cuspidal and residual spectrum. On the other hand, even with the detailed knowledge about the constituents of the local Arthur packets, when a global representation is formed (by choosing members of local packets), it is still difficult to decide whether it is automorphic; a very non-trivial multiplicity formula is involved. In this respect, our method is very direct and gives an explicit realization of an automorphic representation, with lots of information on the local components.

We want to stress that our construction of automorphic representations is unconditional. It explicitly constructs a residual representation attached to an Arthur parameter. This parameter is of special kind (conforming to conditions $(\Delta )$ we describe in the third section), but for such parameter the construction is not making any additional assumptions. It is not applying the trace formula, thus is independent of the still unproven stabilization of the twisted trace formula for $GL_n$, and makes no additional assumptions, such as regular infinitesimal character or cohomological requirements.

By our (inductive) construction, all these representations we obtain (except maybe at the basis of our construction) are residual. A result on classification of residual representations is given in [12], and in [11] are given, by standard procedure of Langlands spectral decomposition, the quadratic unipotent Arthur parameters which parametrize automorphic representations of symplectic group with a non-zero constant term along Borel subgroup.

The idea of using degenerate Eisenstein series to insure the automorphicity of the constructed representation was suggested by Goran Muić in relation with his earlier work [18]. We construct automorphic representations for quite general quadratic unipotent Arthur parameters, and these representations need not to have a non-zero constant term along Borel subgroups (unlike to those in [18]); the inductive construction starts from the unipotent automorphic representation with the general support (i.e., a minimal parabolic subgroup with non-vanishing constant term along that parabolic subgroup). In our inductive construction, to prove the non-vanishing of degenerate Eisenstein series applied to certain global representation (and thus proving that this representation is automorphic) we calculate the constant term of the Eisenstein series in question. In that calculation, a sum of intertwining operators occurs. We use an argument which is essentially the one in [15] (the second chapter, the first section), but since the assumptions are somewhat different than in loc.cit (where representation of which Eisenstein series is being taken is cuspidal, and in our situation, this might not be the case) for the benefit of the reader, we make an explicit calculation (Theorem 5.3).

In the second section, we prove some claims about the structure of local components of the automorphic representations of symplectic and general linear groups related to quadratic unipotent parameters. These claims are not consequences of our constructions; they are quite general and apply to every automorphic square-integrable representation with quadratic unipotent Arthur parameter. Let $\Pi$ be (any) irreducible automorphic representation appearing (discretely) in the space of square-integrable automorphic forms on $Sp_{2n}(\mathbb {A}).$ Then, there exists a standard parabolic subgroup $P(\mathbb {A})$ of $Sp_{2n}(\mathbb {A})$ with the standard Levi subgroup $M(\mathbb {A})$ and a cuspidal automorphic representation $\pi _0$ of $M$ such that, as abstract representations, $\pi$ is embedded in $\mathrm{ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}\pi _0$ (the precise statement of the result and realization in the space of automorphic forms which we need in the fourth chapter are given in Theorem 5.1). We know that a Levi subgroup $M$ is a product of general linear factors with the symplectic group of smaller rank. Now, using some simple consequences of the classification of the unitary unramified representations of classical groups [19] we get that the factors of $\pi _0$ corresponding to the general linear factors are of Ramanujan type [22], and the factor corresponding to the representation of the smaller symplectic group is a cuspidal automorphic representation with almost all local components which are negative representations (the notion is defined further on in the preliminaries section and is essential in understanding of the local components of the automorphic representations attached to quadratic unipotent parameters). The results of the second section can be obtained even without the embedding of Theorem 5.1, i.e., of its abstract-representation counterpart mentioned above-these results would also follow from the global Langlands’ subquotient theorem [6], but we need Theorem 5.1 for the calculations in the fourth section.

In the third section we explain in detail the inductive nature of finding automorphic representations with the prescribed quadratic unipotent Arthur parameter. We explain how, for a given parameter, we find “a basic parameter”, and then how we start with the inductive procedure. This procedure is divided in two steps, Step 1 and Step 2, which we explain. We state the form of the Arthur parameter (Conditions $(\Delta )$) for which this procedure is carried out.

In the fourth section we cover the case in which, in each step, all the intermediate (and final) automorphic representation have a non-zero constant term along the standard Borel subgroup. At the end, to show that the final automorphic square-integrable representation of $Sp_{2n}(\mathbb {A})$ is really corresponding to the given Arthur parameter $\phi ,$ we use local results from the Preliminaries (sub)section (Proposition 1.2).

In the fifth section, using a result mentioned above, about embedding of a non-cuspidal automorphic representation in the representation induced from a cuspidal automorphic representation, we obtain the same results as in the fourth section about construction of the representation with the prescribed quadratic unipotent parameter, but now we do not assume that “a basic representation” has a non-zero constant term along the standard Borel subgroup. In this case, we also have to use some extra ingredients, due to a number of different authors (Arthur, Mœglin, Ban), concerning some properties of local Arthur packets, as is briefly explained at the beginning of the third section.

1.2 Preliminaries

Let $k$ be an algebraic number field and $k_v$ its completion at place $v.$ The symplectic group of rank $n$ over $k$ is defined in the following way:

where $J_n$ is $n\times n$ matrix defined by . In general, if $G$ is an algebraic group over $k, B$ is an $k$-algebra, then $G(B)$ denotes the group of $B$-points of $G.$ The adele ring of $k$ we denote by $\mathbb {A}=\mathbb {A}_k,$ and for a finite place $v$ of $k,$ we denote by $O_v$ the ring of integers of $k_v,$ and by $\overline{\omega }_v$ a generator of it’s maximal ideal (we fix it). We let $|\cdot |_v$ (or $\nu _v$) denote the normalized absolute value on $k_v.$ We choose a non-trivial additive character $\psi :k {\setminus } \mathbb {A}\rightarrow \mathbb {C}^{*};$ we have $\psi =\prod _v ^{'}\psi _v$ (a restricted product; $\psi _v$ is unramified almost everywhere). We fix a Haar measure $\mathrm{d}x=\prod _v^{'}\mathrm{d}x_v$ such that $\mathrm{d}x_v$ self-dual with respect to character $\psi _v$ (meaning self-duality with respect to the Fourier transform). A unitary Grossencharacter $\mu :k^{*}{\setminus } \mathbb {A}^{*}\rightarrow \mathbb {C}^{*}$ can be written as a restricted direct product of local characters $\mu _v:k_v\rightarrow \mathbb {C}^{*};$ in this set up, Tate defined local $L$-function $L(s,\mu _v)$ and $\varepsilon$-factor $\varepsilon (s,\mu _v,\psi _v),$ here $\varepsilon (s,\mu )=\prod \varepsilon (s,\mu _v,\psi _v)$ is independent of $\psi .$ For all places where $\mu _v$ and $\psi _v$ are unramified, $\varepsilon (s,\mu _v,\psi _v)=1.$ Meromorphic continuation of $L(s,\mu )=\prod L(s,\mu _v)$ (the product initially converges for $Re(s)>1$) satisfies the following functional equation

$$\begin{aligned} L(s,\mu )=\varepsilon (s,\mu )L(1-s,\mu ^{-1}). \end{aligned}$$

If $\mu \ne |\cdot |^{it},\; t\in \mathbb {R},\,L(s,\mu )$ is entire. Otherwise, it has simple poles for $s=-it$ and $s=1-it.$ Since we are interested in the cases where $\mu ^2=1,$ to detect the poles, it will be enough only to look at $s\in \mathbb {R}.$ Thus only the case of $\mu =1$ produces a pole of $L(s,\mu ).$

For $Sp_{2n}$ defined above, we fix a Borel subgroup consisting of upper-triangular matrices, and maximal ($k$-split) torus $T_n$ consisting of diagonal matrices in $B_n.$ In this setting, we denote by $U_n$ the unipotent radical of $B_n.$ With this choice of maximal torus and positive roots ($\Sigma ^+$) determined by $B_n,$ the set of simple roots $\Delta =\Delta (Sp_{2n})$ is given by $\alpha _i=e_i-e_{i+1},\;i=1,\ldots ,n-1$ and $\alpha _n=2e_n$ (here $e_i(t)=t_i,$ where $t=diag(t_1,t_2,\ldots ,t_n,t_n^{-1},\ldots ,t_1^{-1})$ is a diagonal matrix belonging to the maximal torus $T_n$). Corresponding coroots are given by $\check{\alpha _i}=\check{e_i}-\check{e_{i+1}},\;\check{\alpha _n}=\check{e_n},$ where cocharacters $\check{e_i}$ have obvious meaning. For any $\alpha \in \Sigma ^+,$ the corresponding root subgroup $U(\mathbb {A})$ is isomorphic to $\mathbb {A};$ similarly, for every place $v,\,U(k_v)\cong k_v.$ Using this isomorphisms, we define Haar measures on these roots subgroups of unipotent radicals, moreover, in this way we fix Haar measures on unipotent radicals of standard parabolic subgroups, since they are spanned by root subgroups (these are isomorphisms of $k$-varieties). The normalization of measures on unipotent radicals is important because of the definition of intertwining operators.

For every $v<\infty$ we fix $K_v=Sp_{2n}(O_{k_v})$ as a maximal good compact subgroup of $Sp_{2n}(k_v),$ and for archimedean places we fix some maximal compact subgroup and denote $K=\prod _vK_v\subset Sp_{2n}(\mathbb {A}).$ Also we denote $K_{\infty }=\prod _{v| \infty }K_ v.$ Let $\mathfrak {g}_{\infty }$ be a (real) Lie algebra of $\prod _{v|\infty }Sp_{2n}(k_v).$ We say that an irreducible admissible representation of $Sp_{2n}(k_v)$ ($v<\infty$) is unramified if it has a non-zero $K_v$-fixed vector (also called $K_v$-spherical). Then, necessarily, this vector is unique, up to a scalar. We have the following important result:

Lemma 1.1

Let $G=Sp_{2n}(k_v)$ and $K_v$ as above. Assume that $\sigma$ is $K_v$-spherical smooth representation of $G,$ and $\sigma$ is a subquotient of $\mathrm{Ind}_{MN}(\sigma '\otimes 1_N),$ for some smooth representation $\sigma '$ of $M$ (where $M$ is a standard Levi subgroup of a standard parabolic subgroup $MN$). Then $\sigma '$ is $M\cap K_v$-spherical.

Proof

This is Lemma 1.1 (ii) of [16]. $\square$

The main aim of this paper is attaching (fairly explicitly) an automorphic representation to a quadratic unipotent Arthur parameter (cf. [1]). So we recall the form of Arthur parameters we need (without invoking the conjectural Langlands group $L_k$). Let $W_k$ denote the global Weil group attached to an algebraic number field $k.$

The general quadratic unipotent Arthur parameter is an admissible homomorphism

$$\begin{aligned} \phi :W_k\times SL(2,\mathbb {C})\rightarrow SO(2n+1;\mathbb {C}) \end{aligned}$$

such that the following holds

• 1. $\phi _{|W_k}$ is continuous, semisimple, and it’s image is bounded,

• 2. $\phi _{|SL(2,\mathbb {C})}$ is a morphism of algebraic groups,

• 3. $\phi (W_k\times SL(2,\mathbb {C}))$ is discrete, that is, the image is not contained in a proper Levi subgroup of $SO(2n+1;\mathbb {C}),$

• 4. $\phi _{|W_k}$ factors through $W_k^{ab},$ where $W_k^{ab}$ is the maximal abelian Hausdorff quotient of $W_k.$

We can thus decompose homomorphism $\phi$ into the sum of irreducible representations of $W_k^{ab}\times SL(2,\mathbb {C})$ so that

$$\begin{aligned} \phi \cong \oplus _{(\mu ,a)\in \mathrm{Jord}(\phi )}\mu \otimes V_a, \end{aligned}$$

where $V_a$ is the unique irreducible algebraic representation of $SL(2,\mathbb {C})$ of dimension $a$ and $\mu :k^{*}{\setminus } \mathbb {A}^{*}\rightarrow \mathbb {C}^{*}$ is a quadratic Grossencharacter of $k,$ obtained by class field theory from a character of $W_k^{ab}.$ This decomposition defines $\mathrm{Jord}(\phi ).$ It can be shown (cf. Theorem 4.1 of [18]) that the set of parameters like this is in the bijective correspondence with the collection of finite sets Jord (called Jordan blocks) consisting of pairs $(\mu ,a)$, where $\mu :k^{*}{\setminus } \mathbb {A}\rightarrow \mathbb {C}^{*}$ is a quadratic Grossencharacter of $k$, $a$ is an odd positive rational integer such that $\sum _{(\mu ,a) \in \mathrm{Jord}}a=2n+1$ and $\prod _{(\mu ,a)\in \mathrm{Jord}}\mu =1.$ So the bijection is $\mathrm{Jord}\Longleftrightarrow \mathrm{Jord}(\phi ).$

To this global Arthur parameter we attach a local parameter:

$$\begin{aligned} \phi _v\cong \oplus _{(\mu ,a)\in \mathrm{Jord}(\phi )}\mu _v\otimes V_a. \end{aligned}$$

Note that, in this (local) case, $\mathrm{Jord}(\phi _v)$ (defined analogously to the global case) does not have to be a set (i.e., for different global $\mu$ and $\lambda$ we can have $(\mu _v,a)=(\lambda _v,a)$).

We say that an irreducible subrepresentation $X\cong \otimes _v X_v$ of the space $A^2(Sp_{2n}(k){\setminus } Sp_{2n}(\mathbb {A}))$ of square-integrable automorphic forms on $Sp_{2n}(\mathbb {A})$ is attached to a global unipotent parameter $\phi ,$ if for all but a finite number of places $v,\,X_v$ is unramified representation attached to parameter $\phi _v$ (so necessarily negative representation; the definition of a (strongly) negative representation of $Sp_{2n}(k_v),\;v<\infty$ and its Arthur parameter (i.e., the Jordan blocks) is recalled below).

In this paper we construct inductively an automorphic representation with the prescribed quadratic unipotent Arthur parameter like this, in addition, we assume the condition ($\Delta$) explained in the beginning of the third section.

For an irreducible representation $\pi$ of $Sp_{2n}(k_v) \;(v <\infty )$, by $r_{1,\ldots ,1;0}(\pi )$ we denote the Jacquet module of that representation with respect to the Borel subgroup $B_n.$ An irreducible admissible unramified representation of $Sp_{2n}(k_v)$ is strictly (or strongly) negative if for every irreducible subquotient $\chi _1\nu ^{s_1}\otimes \cdots \otimes \chi _n\nu ^{s_n}$ of $r_{1,\ldots ,1;0}(\pi )$ (here $\chi _i$ is unitary character of $GL(1,k_v),\; s_i\in \mathbb {R},\;i=1,\ldots ,n$) the following holds

$$\begin{aligned}&s_1<0,\nonumber \\&s_1+s_2<0,\nonumber \\&\qquad \quad \vdots \nonumber \\&s_1+s_2+\cdots +s_n<0. \end{aligned}$$

(1)

In an analogous circumstances, an unramified representation is negative if the inequalities above are not necessarily strict. Note that this criterion is reverse of Casselman’s criterion for square-integrability for the representations of symplectic groups; indeed the Aubert duals of strictly negative (resp. negative) representations are square-integrable (resp. tempered) representations.

We use Zelevinsky notation for the normalized parabolic induction for the general linear groups and the symplectic group. Let $M\cong GL_{n_1}(k_v)\times \cdots \times GL_{n_k}(k_v)$ be a ($k_v$-points of) Levi subgroup of a standard parabolic subgroup $P$ of $GL_n(k_v)$ so that $n_1+\cdots +n_k=n.$ Let $\pi _1,\ldots ,\pi _k$ be admissible representations of $GL_{n_i}(k_v),\; i=1,\ldots ,k.$ Then, we denote $\mathrm{Ind}_P^{GL_n(k_v)}(\pi _1\otimes \cdots \otimes \pi _k)$ by $\pi _1\times \cdots \times \pi _k.$ Similarly, let $M\cong GL_{n_1}(k_v)\times \cdots \times GL_{n_k}(k_v)\times Sp_{2n'}(k_v)$ be a ($k_v$-points of) Levi subgroup of a standard parabolic subgroup $P$ of $Sp_{2n}(k_v)$ so that $n_1+\cdots +n_k+n'=n.$ Let $\pi _1,\ldots ,\pi _k$ be admissible representations of $GL_{n_i}(k_v),\; i=1,\ldots ,k$ and let $\sigma$ be an admissible representation of $Sp_{2n'}(k_v).$ Then, we denote $\mathrm{Ind}_P^{Sp_{2n}(k_v)}(\pi _1\otimes \cdots \otimes \pi _k\otimes \sigma )$ by $\pi _1\times \cdots \times \pi _k\rtimes \sigma .$ If $n'=0$, i.e., $P$ is a subgroup of the Siegel parabolic subgroup, then we denote this induced representation by $\pi _1\times \cdots \times \pi _k\rtimes 1.$ Here $1$ thus denotes the trivial representation of the trivial group.

For $v< \infty ,$ a unitary character $\chi$ of $GL_1(k_v)$, and $\alpha ,\beta \in \mathbb {R}$ such that $\beta +\alpha +1\in \mathbb {Z}_{\ge 1},$ by $\zeta (-\alpha ,\beta ;\chi )$ we denote the unique irreducible subrepresentation of $\nu ^{-\alpha }\chi \times \nu ^{-\alpha +1}\chi \times \cdots \times \nu ^{\beta }\chi$ (which is a character $\nu ^{\frac{\beta -\alpha }{2}}\chi \circ \mathrm{det}$ of $GL_{\beta +\alpha +1}(k_v)$). For $v<\infty ,$ let $\chi _0$ be the unique quadratic non-trivial unramified character of $k_v^{*}.$

The Jordan blocks are defined for an unramified strongly negative and negative representations of a symplectic group $Sp_{2n}(k_v)\; (v<\infty )$ as follows. For an unramified strictly negative representation $\sigma$ of $Sp_{2n}(k_v)$ there exists (a unique) set of positive odd rational integers $2m_1+1<2m_2+1<\cdots <2m_l+1$ and $2n_1+1<2n_2+1<\cdots <2n_k+1$ such that $k$ is even and $2m_1+1+\cdots +2m_l+1+2n_1+1\cdots +2n_k+1=2n+1$ (so $l$ is odd) such that

$$\begin{aligned}&\sigma \hookrightarrow \zeta (-n_{k},n_{k-1};\chi _0)\times \cdots \times \zeta (-n_2,n_1;\chi _0) \nonumber \\&\quad \times \zeta (-m_l,m_{l-1};1)\times \cdots \times \zeta (-m_3,m_2;1)\times \zeta (-m_1,-1;1)\rtimes 1, \end{aligned}$$

(2)

where, if $m_1=0,$ there is no factor $\zeta (-m_1,-1;1)$ (cf. [16, Lemma 5.5]). Then, we define, for $\chi _0$ (the quadratic unramified character defined above) and $1=1_{GL_1(k_v)}$, the trivial character of $GL_1(k_v),$ the following set, called the Jordan block of $\sigma :$

$$\begin{aligned} \mathrm{Jord}(\sigma )=\{(\chi _0,2n_1+1),\ldots ,(\chi _0,2n_k+1),(1,2m_1+1),\ldots ,(1,2m_l+1)\}. \end{aligned}$$

For a negative representation $\sigma _n$ there exists a unique strongly negative representation $\sigma _{sn}$ and pairs $(\chi _1,l_1),\ldots ,(\chi _j,l_j)$ ($l_i\in \mathbb {Z}_{\ge 1},\,\chi _i$ unramified unitary characters) unique up to a permutation and taking inverses of characters, such that (cf. [19, Theorem 0-3])

$$\begin{aligned} \sigma _n \hookrightarrow \times _{i=1}^j\zeta \left( -\frac{l_i-1}{2},\frac{l_i-1}{2};\chi _i\right) \rtimes \sigma _{sn}. \end{aligned}$$

Then we define a multiset $\mathrm{Jord}(\sigma _n)=\mathrm{Jord}(\sigma _{sn})+\sum _{i=1}^k\{(\chi _i,l_i),(\chi _i^{-1},l_i)\}.$

Assume $\phi$ is an Arthur parameter as above. Then, for a global quadratic Grossencharacter $\chi ,$ we denote $\mathrm{Jord}_{\chi }(\phi )= \{a: (\chi ,a)\in \mathrm{Jord}(\phi )\}.$ In the same way, for a local component $\phi _v$ of an Arthur parameter and local character $\chi _v,$ we define $\mathrm{Jord}_{\chi _v}(\phi _v).$ Analogously, for a (local) negative representation $\sigma$ and a local character $\chi ,$ we define $\mathrm{Jord}_{\chi }(\sigma )=\{a: (\chi ,a)\in \mathrm{Jord}(\sigma )\}.$

What we need in future from this information about negative representation is the following result.

Proposition 1.2

Let $\beta >\alpha >0$ be integers, and let $\chi$ be an unramified quadratic character of $k_v.$ Let $\pi$ be a negative representation. Then, the irreducible unramified subquotient of $\zeta (-\beta ,\alpha ;\chi )\rtimes \pi$ is a subrepresentation; it is also a negative representation.

Proof

This is the main result in [9].$\square$

2 Restriction on the local components

In this section we comment on some information about local components of automorphic representations of symplectic group attached to unipotent Arthur parameter and some global consequences that we can get relatively straightforward from the classification of spherical representations for general linear and symplectic groups. These consequences we relate to generalized Ramanujan conjecture (cf. [22]).

Firstly, for the convenience of reader, we recall the classification of unitary unramified representations for $p$-adic general linear groups (cf. [19, Theorem 4-1]). By $\mathrm{Irr}^{unr}(GL)$ we denote a set of the equivalence classes of irreducible unramified representations for the general linear groups $GL_n(k_v)$ for all $n\in \mathbb {Z}_{\ge 1}.$ Here $\mathrm{Irr}^{u,unr}(GL)$ denotes a subset of $\mathrm{Irr}^{unr}(GL)$ consisting of all the classes of unitary unramified representations.

Theorem 2.1

• (i) Let $\phi _1,\ldots ,\phi _{a},\psi _1,\ldots ,\psi _b \in \mathrm{Irr}^{unr}(GL)$ be a sequence of unramified unitary characters (one-dimensional unramified representations). Let $\alpha _1,\ldots , \alpha _b\in \langle 0, \frac{1}{2} \rangle$ be a sequence of real numbers (the possibility of $a=0$ or $b=0$ is not excluded here). Then

  $$\begin{aligned} \phi _1\times \cdots \times \phi _a\times (\nu ^{\alpha _1}\psi _1\times \nu ^{-\alpha _1}\psi _1)\times \cdots \times (\nu ^{\alpha _b}\psi _b\times \nu ^{-\alpha _b}\psi _b)\in \mathrm{Irr}^{u,unr}(GL). \end{aligned}$$

• (ii) Let $\pi \in \mathrm{Irr}^{u,unr}(GL).$ Then there exist $\phi _1,\ldots ,\phi _{a},\psi _1,\ldots ,\psi _b$ and $\alpha _1,\ldots , \alpha _b$ as in (i) such that $\pi$ is isomorphic to the induced representation given above. Each sequence $\phi _1,\ldots ,\phi _{a}$ and $(\psi _1,\alpha _1),\ldots , (\psi _b,\alpha _b)$ is uniquely determined by $\pi$ up to a permutation.

Now we return to automorphic representation $\Pi$ of $Sp_{2n}(\mathbb {A})$ (more precisely, of $\prod _{v<\infty }Sp_{2n}(k_v)\times (\mathfrak {g}_{\infty },K_{\infty }))$ in square-integrable discrete spectrum of $Sp_{2n}(\mathbb {A})$ attached to an unipotent Arthur parameter. In [10] we prove that there is an embedding (as abstract representations)

$$\begin{aligned} \Pi \hookrightarrow \pi _1\times \cdots \pi _k\rtimes \sigma , \end{aligned}$$

(3)

where $\pi _1,\ldots ,\pi _k$ are (essentially) cuspidal automorphic representations of appropriate general linear groups, $\sigma$ is an automorphic cuspidal representation of $Sp_{2m}(\mathbb {A}).$

We know that, at almost all places $v,\,\Pi _v$ is an unramified negative representation (with the cuspidal support expressible in terms of quadratic characters entering the expression for the local Arthur parameter). At these places we look at the local embedding

$$\begin{aligned} \Pi _v \hookrightarrow \pi _{1,v}\times \cdots \pi _{k,v}\rtimes \sigma _v. \end{aligned}$$

(4)

By Lemma 1.1, we know that $\pi _{1,v},\ldots , \pi _{k,v},\sigma _v$ are spherical representations. By the main result of [19, (Theorem 0.8)] $\sigma _v,$ as a unitary unramified representation of $Sp_{2m}(k_v),$ is irreducibly induced

$$\begin{aligned} \sigma _v\cong \times _{(l,\chi ,\alpha )} \left( \zeta \left( -\frac{l-1}{2},\frac{l-1}{2};1_v\right) \nu ^{\alpha }\chi \right) \rtimes \sigma _{v,neg}. \end{aligned}$$

(5)

Here $(l,\chi ,\alpha )$ varies through members of certain finite set, $\chi$ is unramified unitary character (but we know that cuspidal support of $\sigma _v$ consists of quadratic characters, so $\chi \in \{\chi _{0,v},1_v\}$); $l$ is positive rational integer, and $\alpha \in \langle 0,1 \rangle ,$ satisfying some technical conditions. Here $\chi _{0,v}$ denotes the unique unramified non-trivial quadratic character of $k_v^{*}$ and $1_v$ is the trivial character of $k_v^{*}.$ The representation $\sigma _{v,neg}$ is negative. Since the exponents in the cuspidal support of $\sigma _v$ must be rational integers, in the above expression we must have $\frac{l-1}{2}+\alpha \in \mathbb {Z},$ and $-\frac{l-1}{2}+\alpha \in \mathbb {Z},$ so $\alpha =\frac{1}{2}$ and $l$ is even. Having this in mind, we examine some of these technical conditions mentioned above (cf. Definition 0–7 of [19]); namely we must apply Definition 0–7 (3) of [19], where, for fixed $l$ and $\chi ,$ exponents in $e(l,\chi )$ are collected. Here $e(l,\chi )$ denotes all the $\alpha$’s which occur in (5) for fixed $l$ and $\chi .$ We thus have only one exponent, $\alpha =\frac{1}{2}.$ Then, using notation from loc.cit, we have that $v=0.$ On the other hand, from (3)(b) it follows that $u=1.$ So, $u+v$ is odd, meaning, by 3(a), that $(\chi ,l) \in \mathrm{Jord}(\sigma _{v,neg}),$ for every $(\chi ,l)$ appearing in (5). We know ([16], [19, Theorem 0-3]) that, for $\sigma _{v,neg}$ there exists $\sigma _{v,sn},$ a strongly negative representation of some $Sp_{2s_0}(k_v),$ and unitary unramified characters $\chi _1,\ldots ,\chi _r$ and $l_1,\ldots ,l_r \in \mathbb {Z}_{\ge 1}$ such that

$$\begin{aligned} \sigma _{v,neg}\hookrightarrow \times _{j=1}^r \zeta \left( -\frac{l_j-1}{2},\frac{l_j-1}{2};\chi _j\right) \rtimes \sigma _{v,sn}, \end{aligned}$$

(6)

where, in our situation, again $\chi _j\in \{\chi _{0,v},1_v\}$ and $\mathrm{Jord}(\sigma _{v,neg})=\mathrm{Jord}(\sigma _{v,sn}) \cup \sum _{j=1}^r\{(\chi _j,l_j),(\chi _j^{-1},l_j^{-1})\}.$ Since we are in the symplectic group case, every number $l$ in $(l,\chi )\in \mathrm{Jord}(\sigma _{v,sn})$ is odd. This means that every $l$ in $(l,\frac{1}{2})$ from (5) must be one of $(\chi _j,l_j)$ in (6). But, since such $l$ is even, this would, by (6), mean that there is an exponent in the cuspidal support of $\sigma _{v,neg},$ so in the cuspidal support of $\Pi _v,$ which is not an integer; a contradiction. This means that (5) reduces to $\sigma _v=\sigma _{v,neg}$ is a negative representation.

Now we analyze spherical representations of general linear group in (4). We note that, for every $j=1,\ldots ,k$, there exists a real number $\beta _j$ such that $\nu ^{-\beta _j}\pi _j$ is a cuspidal (unitary) representation of some global general linear group (cf. 3). For the simplicity of notation, assume $j=1.$ Then,

$$\begin{aligned} \nu ^{-\beta _1}\pi _{1,v}\cong \phi _1\times \cdots \times \phi _a\times (\nu ^{\alpha _1}\psi _1\times \nu ^{-\alpha _1}\psi _1)\times \cdots \times (\nu ^{\alpha _b}\psi _b\times \nu ^{-\alpha _b}\psi _b) \end{aligned}$$

(Theorem 2.1). Here $\phi _j=\zeta (-\frac{t-1}{2},\frac{t-1}{2};\phi _j),$ i.e., we look at $\phi _j$ as a character of $GL_t(k_v);$ analogously for $\psi _j.$ Note that $\nu ^{-\beta _1}\pi _{1,v}$ has to be a generic representation, so every factor in the previous relation must be generic, forcing $\phi$’s and $\psi$’s to be characters of $GL_1(k_v).$ Note that then $\beta _1+\alpha _j$ and $\beta _1-\alpha _j$ have to be integers, for $j=1,\ldots ,b.$ By subtracting these expressions, we get that $2\alpha _j\in \mathbb {Z},$ which is impossible (by assumptions in Theorem 2.1). This means that $\nu ^{-\beta _1}\pi _{1,v}\cong \phi _1\times \cdots \times \phi _a,$ where $\phi _j$’s are unramified (quadratic) characters of $GL_1(k_v).$

Keeping the notation from (3) and (4) we have proved the following theorem:

Theorem 2.2

Let $\Pi$ be an irreducible automorphic representation in the discrete (square-integrable) spectrum of $Sp_{2n}(\mathbb {A})$ attached to an unipotent Arthur parameter, so that, at almost all places, $\Pi _v$ is an unramified negative representation.

For those places $v,$ the local components $\sigma _v$ of an automorphic cuspidal representation $\sigma$ such that (3) holds, are negative (unramified) representations. Also, for those places, the local components of a cuspidal automorphic (unitary) representation of $\nu ^{-\beta _j}\pi _j$ of a general linear group are fully induced from quadratic characters. This moreover means, that, at those places, the local components are tempered, i.e., every representation $\nu ^{-\beta _j}\pi _j,\; j=1,\ldots , k$ is of Ramanujan type (cf. [22] and conjectures in the sixth section there).

Remark

Note that for this Theorem we do not actually need the global embedding of (3); it is enough to just use Langlands’ “subquotient version” [6]; we though do need (3) in the fifth section.

We now study a certain specific case of (3). Assume that all the representations $\pi _1,\ldots ,\pi _k$ are global characters of $GL_1(\mathbb {A}),$ denote them $\nu ^{\beta _1}\chi _1,\ldots ,\nu ^{\beta _k}\chi _{k},$ where $\chi$’s a unitary, and $\beta$’s are real numbers. From the fact that almost everywhere the characters $\chi _{i,v},\; i=1,\ldots k,$ are quadratic, it follows that they are quadratic (globally).

Definition 2.1

For a strongly negative unramified representation $\sigma$ or for a negative unramified $\sigma$ whose cuspidal support consists only of quadratic characters, by $\mathrm{supp}(\mathrm{Jord}(\sigma ))$ we denote the multiset

$$\begin{aligned} \mathrm{supp}(\mathrm{Jord}(\sigma ))=\sum _{(\chi ,a)\in \mathrm{Jord}(\sigma )}[\nu ^{- \frac{a-1}{2}}\chi ,\nu ^{\frac{a-1}{2}}\chi ]. \end{aligned}$$

Here $[\nu ^{- \frac{a-1}{2}}\chi ,\nu ^{\frac{a-1}{2}}\chi ]$ denotes the set $\{\nu ^{- \frac{a-1}{2}}\chi ,\nu ^{- \frac{a-1}{2}+1}\chi ,\ldots ,\nu ^{\frac{a-1}{2}}\chi \}.$

In the fourth section we need the following:

Corollary 2.3

In the above situation, we have

$$\begin{aligned} \mathrm{supp}(\mathrm{Jord}(\Pi _v))=\mathrm{supp}(\mathrm{Jord}(\sigma _v))+\sum _{i=1}^k\{\chi _{i,v}\nu ^{\beta _i},\chi _{i,v}\nu ^{-\beta _i}\}. \end{aligned}$$

Proof

Since $\Pi _v$ and $\sigma _v$ are unramified negative representations and $\chi _{i,v}$ quadratic, this is obvious.$\square$

3 Description of the inductive construction

The method presented in this paper is, as we have explained in the Introduction, of the inductive nature, the important question is how to construct the first automorphic representation explicitly realized in the space of square-integrable automorphic forms, which corresponds to the part of the given Arthur parameter (i.e., which corresponds to the analogous Arthur parameter for the symplectic group of smaller rank). We now discuss how to see what the “basis” Arthur parameter should look like.

Let $\mathrm{Jord}_\mu (\phi )=\{a:(\mu ,a)\in \mathrm{Jord}(\phi )\}.$ We denote different grossencharacters appearing in the parameter $\phi$ by $\mu _1,\ldots ,\mu _k$ and we denote $|\mathrm{Jord_{\mu _i}}(\phi )|=r_i.$ From the conditions in the definition of Arthur parameter it follows that there is an odd number of different $\mu _i$’s such that $r_i$ is odd. We let $\mathrm{Jord}_{\mu _i}(\phi )=\{a_{i1},\ldots ,a_{ir_i}\},$ where $a_{i1}<a_{i_2}<\cdots <a_{ir_i}.$

We let exactly the first $l$ characters ($l$ is odd) be such that their $r_i$ is odd. It follows that $\prod _i^l\mu _i=1.$ Now, we denote by $\phi _0=\mu _1\otimes V_{a_{11}}\oplus \mu _2\otimes V_{a_{21}}\oplus \cdots \oplus \mu _l\otimes V_{a_{l1}}$ an Arthur parameter of an automorphic representation $\sigma (\phi _0)$ (realized directly in the space of the square-integrable automorphic forms). This representation will be the basis of our inductive procedure.

If $l=1,$ we get $\mu _1=1.$ In that case we know that the trivial representation of $Sp_{a_{11}-1}(\mathbb {A})$ is attached to the Arthur parameter $\phi _0=\mathbf{1}\otimes V_{a_{11}};$ and we know that this representation is automorphic (realized on the space of constant functions on $Sp_{a_{11}-1}(\mathbb {A})$); moreover it belongs to the residual part of the spectrum. Moreover, Kim and Shahidi [11] proved that, among the automorphic square-integrable representations with the unipotent Arthur parameters, exactly these (i.e., the ones with $l=1$) are the ones for which some constant term along the Borel subgroup does not vanish; this means (as can be seen, for example, in [18]) that these representations can be realized (as abstract representations) as subrepresentations of some (global) principal series representations.

We assume that we can find $\sigma (\phi _0),$ even when $l>1.$ There are some instances for which we know, more or less explicitly, representations attached to a parameter of this form; e.g., a cuspidal representation of $SL(2,\mathbb {A})$ attached to the Gelbart–Jacquet lift of a cuspidal representation of $GL(2,\mathbb {A})$ to $GL(3,\mathbb {A})$. The precise statement can be found in [8, the third section]. The case of special interest in this context is a cuspidal representation of $SL(2,\mathbb {A})$ with the parameter $\chi \otimes 1\oplus \mu \otimes 1 \oplus \chi \mu \otimes 1,$ where $\chi ,\mu$ are non-trivial global quadratic characters (related in a way described in [8]. Some of the examples can be obtained by functoriality (from symplectic to general linear group).

To get Arthur parameter $\phi$ from the Arthur parameter $\phi _0$ we have to add an even number of elements from $\mathrm{Jord_{\mu _i}}(\phi ),\;i=1,\ldots ,l$ and whole sets of elements of $\mathrm{Jord_{\mu _i}}(\phi ),\;i=l+1,\ldots ,k$ (every set from this collection has even number of elements).

So assume that we have already constructed some automorphic representation $\sigma (\phi ')$ attached to the parameter $\phi '$ such that $\phi '$ is obtained by adjoining a $\mu _t\otimes V_{a_{t,j}}\oplus \mu _t\otimes V_{a_{t,j+1}}$ in each step (we of course, have a finite number of steps). Here, if $t\in \{1,\ldots ,l\}$ we start with $\mu _t\otimes V_{a_{t2}}\oplus \mu _t\otimes V_{a_{t3}},$ and when we have $t\in \{l+1,\ldots ,k\}$ we start with $\mu _t\otimes V_{a_{t1}}\oplus \mu _t\otimes V_{a_{t2}}.$ So, we assume that we have an explicit realization of $\sigma (\phi '),$ (one of) the automorphic representations with the Arthur parameter $\phi '$, inside the space of square integrable automorphic forms $A_2(Sp_{2n'}(k){\setminus } Sp_{2n'}(\mathbb {A})).$

We want to construct explicitly (i.e., realize explicitly in the space of the square-integrable automorphic forms) an automorphic representation with the Arthur parameter $\phi =\phi '\oplus \mu _t\otimes V_{a_{t,j}}\oplus \mu _t\otimes V_{a_{t,j+1}}$ (so that $\mu _t$ may or may not appear in $\phi ';$ if $\mu _t\otimes V_b$ appears in $\phi '$ then $b<a_{t,j}$).

To be able to perform our inductive procedure, we assume that our quadratic unipotent parameter is of such a form that the addition of new elements in the Jordan block can be arranged to satisfy the following

Conditions $(\Delta )$

• 1. Each of the numbers $a_{11},\ldots ,a_{l1}$ entering the definition of $\phi _0$ is strictly smaller than any element $a_{t,j}$ we subsequently add in the Jordan block

• 2. We add elements $a_{t,j}$ and $a_{t,j+1}$ in the Jordan block in such an order that $a_{t,j}$ is equal or greater to any other element added previously in the Jordan block (this means that we do not necessarily add all the elements of $\mathrm{Jord}_{\mu _1}$ and then all the elements of $\mathrm{Jord}_{\mu _2}$ and so on).

The method we use (for both cases, $l=1$ and $l>1$) is the global analog of the construction of discrete series of classical $p$-adic groups [14]; this (global) method is already present in [18, (cf. Theorem 5.2)]. We briefly describe it. It consists of two steps:

Step 1

We first form a global representation of $Sp_{2n'+2a_{t,j}}(\mathbb {A})$ induced from a representation $|\mathrm{det}|^s\mu _{t} \otimes \sigma (\phi ')$ of $GL(a_{t,j},\mathbb {A})\otimes Sp_{2n'}(\mathbb {A}).$ In our realization of this representation in a certain space of automorphic forms, we can get Eisenstein series (as an intertwining operator) acting on it. This Eisenstein series is holomorphic at $s=0.$ We will then describe a certain subrepresentation $\pi$ of this space (for $s=0$) on which this Eisenstein series will be non-zero, giving our subrepresentation a different realization $E_0(\pi )$ (we use a constant term along the appropriate parabolic subgroup to detect a suitable subrepresentation).

Step 2

Now, we again form a global representation of $Sp_{2n}(\mathbb {A})$ (where $n=n'+\frac{a_{t,j}+a_{t,j+1}}{2}$) induced from a representation $|\mathrm{det}|^s\mu _t\otimes E_0(\pi )$ of $GL(\frac{a_{t,j+1}-a_{t,j}}{2},\mathbb {A})\otimes Sp_{2n'+2a_{t,j}}(\mathbb {A}).$ Again, in the appropriate realization, we can take an Eisenstein series acting on this representation. We prove that this Eisenstein series has a pole of order two for $s= \frac{a_{t,j}+a_{t,j+1}}{4}.$ Then, when we normalize the Eisenstein series with $(s-\frac{a_{t,j}+a_{t,j+1}}{4})^2,$ we get an image representation in the space of square-integrable automorphic forms, with the local structure which relatively easy to analyze for almost every finite place. Now, using our local results from the second section, we see that almost everywhere, the local image is an irreducible negative representation, and we obtain an automorphic representation with the Arthur parameter $\phi .$

We see that this construction is the global analogue of the local construction of the discrete series representations [14], so that the first step corresponds to the formation of the tempered representations, and the second step gives us (the new) discrete series representations.

In the next section we prove all the intermediate steps described above for the situation when $l=1;$ in the fifth section we prove the analogous claims for the parameters with $l>1$ (but sometimes we slightly change our approach, e.g., in the proof of Proposition 5.7). To do so, we need some additional arguments (the embedding of the global automorphic representation in the representation induced from the automorphic cuspidal representation of a Levi subgroup [10], explicit description of the calculation of the constant term of Eisenstein series in a general situation (Theorem 5.3), analogous to a similar result in [15]; we also need some results of Arthur on the structure (with respect to Langlands parameters) of the representations in Arthur packets [1] and of Mœglin about the action of the (Aubert)-duality operator on the representations in the Arthur’s packets [1, 13].

4 Explicit construction I: representations supported in the Borel subgroup

If the representation $\sigma (\phi _0)$ (realized on the $\prod _{v<\infty }Sp_{2n_0}(k_v)\times (\mathfrak {g}_{\infty },K_{\infty })$-invariant irreducible subspace of $A(Sp_{2n_0}(k){\setminus } Sp_{2n_0}(\mathbb {A})$) is concentrated on the Borel subgroup (i.e., we are in $l=1$ case, as explained in the beginning of the third section), then from the proof of Lemma I.3.2 of [15] (as is explained in the third section of [18]), by taking the constant term of $\sigma (\phi _0)$ along the Borel subgroup, we get intertwining which realizes our representation inside the globally induced representation (normalized induction, $K$-finite vectors) of $Sp_{2n_0}(\mathbb {A}),$ induced from a character of the (standard) Borel subgroup (cf. also section 2 of [18]). Because we want an automorphic discrete series representation $\sigma (\phi _0)$ to be attached to the parameter $\phi _0,$ by looking at the corresponding local Arthur parameters we get that it is embedded (i.e., can be realized), as explained above, into the global representation induced from the character $\lambda (\phi _0),$ where

$$\begin{aligned} \lambda (\phi _0)=\nu ^{-\frac{a_{1,1}-1}{2}}\otimes \cdots \otimes \nu ^{-1} \end{aligned}$$

is a character of a maximal split torus of $Sp_{2n_0}(\mathbb {A})$ (so $2n_0=a_{1,1}-1$). This means that $\sigma (\phi ')$ is embedded in the global representation induced from the character $\lambda (\phi ')$

$$\begin{aligned} \lambda (\phi ')&= (|\cdot |^{-\frac{a_{i,l}-1}{2}}\mu _i\otimes |\cdot |^{-\frac{a_{i,l}-1}{2}+1}\mu _i\otimes \cdots \otimes |\cdot |^{\frac{a_{i,l-1}-1}{2}}\mu _i)\otimes \cdots \nonumber \\&\otimes |\cdot |^{-\frac{a_{1,1}-1}{2}}\otimes \cdots \otimes |\cdot |^{-1}. \end{aligned}$$

(7)

Here $\mu _i\in \{\mu _1,\ldots ,\mu _k\}$ and $l\in \{2,\ldots ,r_i\}.$ Here, we assume that, inductively, we have proved, that in each adding of two elements $(\chi ,\alpha )$ and $(\chi ,\beta )$ (where $\alpha <\beta$ are odd positive integers; and we add elements in $\mathrm{Jord}_{\chi }$ by increasing order) we get that our two-step procedure, in a case of descripting this character of a maximal torus, boils down to adding $|\cdot |^{-\frac{\beta -1}{2}}\chi \otimes \cdots \otimes |\cdot |^{\frac{\alpha -1}{2}}\chi$ in front of the previous character (of a smaller torus). Only the character of a basic torus ($|\cdot |^{-\frac{a_{1,1}-1}{2}}\otimes \cdots \otimes |\cdot |^{-1}$) is different, corresponding to the embedding of the trivial representation (global).

4.1 Step 1

We return to the (first step of the) proof. We consider the global induced representation

$$\begin{aligned} \mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi ')), \end{aligned}$$

(8)

where $P$ is a standard parabolic subgroup with Levi subgroup isomorphic to $GL(a_{t,j})\otimes Sp_{2n'}$ (this representation is realized in the analogous way as in [2, p. 32]). Then, the degenerate Eisenstein series, for a section $f_s$ belonging to (8), is given by

$$\begin{aligned} E(s,f_s)(g)=\sum _{P(k){\setminus } Sp_{2n'+2a_{t,j}}(k)} f_s(\gamma g) \end{aligned}$$

is acting on this representation. It is holomorphic at $s=0$ (Langlands, c.f. [2, Theorem 7.2]) and defines an intertwining operator

$$\begin{aligned} \mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi '))\rightarrow A(Sp_{2n'+2a_{t,j}}(k)){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A})). \end{aligned}$$

(9)

Here we assume that $\sigma (\phi ')$ is realized in the space of square-integrable automorphic forms in $A(Sp_{2n'}(k){\setminus } Sp_{2n'}(\mathbb {A})).$

Let $Y$ be a subrepresentation of (8) such that the Eisenstein series does not send it to zero. We want to detect what is a form of such representation. To do so, we calculate the constant term along the Borel subgroup of $E(f_s,\cdot )|_{s=0},\; f_s \in Y,$ and examine when it is non-zero. Since $E(f_s,\cdot )|_{s=0}Y\subset A(Sp_{2n'+2a_{t,j}}(k)){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A})),$ taking the constant term along the appropriate Borel subgroup will give us automatically realization of $E(f_s,\cdot )|_{s=0}Y$ (i.e., of $Y$ since it is irreducible) as a subrepresentation of a representation globally induced from the character of the appropriate maximal torus. So, if we put this together, we calculate $E_0(s,f_s)(g).$ We denote

$$\begin{aligned} \lambda _{\phi ',s,a_{t,j}}=\mu _t|\cdot |^{s-\frac{a_{t,j}-1}{2}}\otimes \cdots \otimes \mu _t|\cdot |^{s+\frac{a_{t,j}-1}{2}}\otimes \lambda (\phi '), \end{aligned}$$

and let $\alpha$ be a simple root such that the parabolic subgroup appearing in (8) corresponds to the set of roots $\Delta {\setminus } \{\alpha \},$ i.e., $\alpha =\alpha _{a_{t,j}}.$ We have

$$\begin{aligned} E_0(s,f_s)(g)&= \int _{U_{n'+a_{t,j}}(k){\setminus } U_{n'+a_{t,j}}(\mathbb {A})}E(f_s,ug)\mathrm{d}u\nonumber \\&= \quad \sum _{w\in W;w(\Delta {\setminus } \{\alpha \})>0}M(\lambda _{\phi ',s,a_{t,j}},w)(f_s)_0(g),\end{aligned}$$

(10)

$$\begin{aligned} (f_s)_0(g)&= \int _{U_{n'}(k){\setminus } U_{n'}(\mathbb {A})}f_s(u'g)\mathrm{d}u'; \end{aligned}$$

(11)

the last integral is actually a canonical map (for $g=k\in K$ it does not depend on $s$) from the automorphic realization of $\sigma (\phi ')$ to the space of it’s constant terms along $B_{n'},$ i.e., embedding into the global representation induced from the character $\lambda (\phi ')$ of the maximal torus $T_{n'}(k){\setminus } T_{n'}(\mathbb {A}).$

This formula for the calculation of the constant term of the Eisenstein series of the representation (8) is proved in Lemma 2.1 of [17] (c.f. Lemma 2.2 there).

We further have

$$\begin{aligned}&\sum _{w\in W;w(\Delta {\setminus } \{\alpha \})>0}M(\lambda _{\phi ',s,a_{t,j}},w)(f_s)_0=(f_s)_0+M(\lambda _{\phi ',s,a_{t,j}},w_0)(f_s)_0\nonumber \\&\quad +\sum _{w\in W;w(\Delta {\setminus } \{\alpha \})>0;w\ne w_0,1}M(\lambda _{\phi ',s,a_{t,j}},w)(f_s)_0, \end{aligned}$$

(12)

where $w_0=w_{l,\Delta }w_{l,\Delta {\setminus } \{\alpha \}}^{-1}$ is the longest element in the Weyl group for $\Sigma ^+$ modulo the longest one in $\Sigma _{\Delta {\setminus } \{\alpha \}}^+.$

Now we examine more thoroughly the decomposition of the global intertwining operator $M(\lambda _{\phi ',s,a_{t,j}},w_0)$ into the local ones.

We decompose the representation (8) into the local components

$$\begin{aligned}&\mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi '))\nonumber \\&\quad \cong \otimes _v\mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi ')). \end{aligned}$$

(13)

We are interested how the local components of (8) decompose for $s=0$ (so we get finite length, unitary representations for $s=0$). To study this decomposition, we need local intertwining operators

$$\begin{aligned}&A(s,|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0):\mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '))\nonumber \\&\quad \rightarrow \mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(|\mathrm{det}|_v^{-s}\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi ')) \end{aligned}$$

defined, standardly, by

$$\begin{aligned} A(s,|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)f_v(g_v)=\int _{N(k_v)}f_v(w_0^{-1}n_vg_v)\mathrm{d}n_v. \end{aligned}$$

$w_0$ in the defining integral is the fixed representative of the Weyl group element $w_0$ (fixed like in [21]). These operators converge for $Re(s)>>0,$ and admit meromorphic continuation on the whole complex plane. Since (11) gives (global) embedding into the representation induced from the character of the maximal torus, this also happens locally, so that we have embedding

$$\begin{aligned} \sigma _v(\phi ')\hookrightarrow \lambda _v(\phi '), \end{aligned}$$

and

$$\begin{aligned} \mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{tj}}(k_v)}(|\mathrm{det}|_v^s\mu _{t,GL(a_{tj},v)}\otimes \sigma _v(\phi '))\hookrightarrow \mathrm{Ind}_{B_{n'+a_{tj}(k_v)}}^{Sp_{2n'+2a_{tj}}(k_v)}(\lambda _{\phi ',s,a_{t,j},v}). \end{aligned}$$

Then, the above intertwining operator $A(s,|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)$ is just the restriction of the intertwining operator $A(\lambda _{\phi ',s,a_{t,j},v})$ acting on

$$\begin{aligned}&\mathrm{Ind}_{B_{n'+a_{t,j}(k_v)}}^{Sp_{2n'+2a_{t,j}}(k_v)}(\lambda _{\phi ',s,a_{t,j},v})\rightarrow \mathrm{Ind}_{B_{n'+a_{t,j}(k_v)}}^{Sp_{2n'+2a_{t,j}}(k_v)}(w_0(\lambda _{\phi ',s,a_{t,j},v})). \end{aligned}$$

To normalize intertwining operators $A(s,|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)$ we use the same normalization as we use (standardly) for the intertwining operators $A(\lambda _{\phi ',s,a_{t,j},v},w_0)$ which act on the principal series representations, so the normalization is by well understood $L$-functions:

$$\begin{aligned}&N(|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)\nonumber \\&\quad =r(\lambda _{\phi ',s,a_{t,j},v},w_0) A(s,|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0), \end{aligned}$$

where, for $\underline{s}=(s_1,\ldots ,s_k)\in \mathbb {C}^k$ and a general local character $\lambda (\underline{s})_v=|\cdot |_v^{s_1}\lambda _{1,v}\otimes \cdots \otimes |\cdot |_v^{s_k}\lambda _{k,v}$ of the maximal torus of the appropriate size, and $w$ from the Weyl group attached to this torus, we have:

$$\begin{aligned} r(\lambda (\underline{s})_v,w)=\prod _{\alpha \in \sum ^{+}, w(\alpha )<0} \frac{L(1,\lambda (\underline{s})_v\circ \check{\alpha })\varepsilon (0,\lambda (\underline{s})_v \circ \check{\alpha },\psi _v)}{L(0,\lambda (\underline{s})_v\circ \check{\alpha })}. \end{aligned}$$

This might not be a standard normalization, but it satisfies usual properties with respect to taking hermitian contragredient, etc. (as is discussed in [18, relations (5-17), (5-18), (5-19)]); moreover, for $s=0,\,N(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)$ is holomorphic and unitary (because the analogous claim holds for $N(\lambda _{\phi ',s,a_{t,j},v},w_0),$ cf. Theorem 2.5 of [18]) and $N(0,\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)^2=Id.$ We use this normalized intertwining operators to define subspaces $Y_v^{\pm }\hookrightarrow \mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '))$ as follows:

$$\begin{aligned}&Y_v(\sigma (\phi '))^{\pm } =\{f\in \mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi ')):\\&\quad \times N(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)f=\pm f.\} \end{aligned}$$

Moreover

$$\begin{aligned} \mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '))=Y_v(\sigma (\phi '))^{+}\oplus Y_v(\sigma (\phi '))^{-}. \end{aligned}$$

Note that if $\sigma (\phi ')_v$ is spherical and $\mu _{t,GL(a_{t,j},k_v)}$ unramified quadratic character, then $Y_v(\sigma (\phi '))^{+}\ne \{0\},$ since it necessarily contains a $K_v$-fixed vector, by the property (iv) of Theorem 2.5 in [18]. If, additionally, $v$ is non-archimedean, then (spherical) representation $Y_v(\sigma (\phi '))^{+}$ is irreducible (and so is $Y_v(\sigma (\phi '))^{-}$) (we can see by working little bit with [16, Corollary 5.1] or directly, when we apply Aubert involution).

So we have

$$\begin{aligned} \mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi '))\cong \oplus _S[\otimes _{v\notin S}Y_v(\sigma (\phi '))^+\otimes _{v\in S}Y_v(\sigma (\phi '))^-], \end{aligned}$$

where $S$ ranges over finite subsets of the set of places of $k.$

Claim

If $|S|$ is even and $\pi$ an irreducible subrepresentation of

$$\begin{aligned} \otimes _{v\notin S}Y_v(\sigma (\phi '))^+\otimes _{v\in S}Y_v(\sigma (\phi '))^{-}, \end{aligned}$$

then (9) is non-trivial on $\pi .$

We now proceed to prove this claim.

We return to (12). Let $f_s$ be a function in (8), so that $(f_s)_0$ is factorizable function in $\mathrm{Ind}_{B_{n'+a_{t,j}(\mathbb {A})}}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(\lambda _{\phi ',s,a_{t,j}}).$ We assume that $(f_s)_0=\otimes _vf_{s,v}$ and for $v\notin S$ $f_{s,v}$ is spherical vector normalized so that $f_{s,v}(e)=1.$ To simplify the notation, we denote the action of the Weyl group on $\lambda _{\phi ',s,a_{t,j}}$ by $w(s)$ (instead of $w(\lambda _{\phi ',s,a_{t,j}})$). Then, for the choice of $\pi$ as in the claim, (12) becomes

$$\begin{aligned}&E_0(s,f_s)(g)=(f_s)_0 +r(\lambda _{\phi ',s,a_{t,j},v},w_0)^{-1}\otimes _{v\notin S}f_{w_0(s),v}\nonumber \\&\quad \otimes _{v\in S} N(|det|^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)f_{s,v}\nonumber \\&\quad +\sum _{w\in W;w(\Delta {\setminus } \{\alpha \})>0;w\ne w_0,1}M(\lambda _{\phi ',s,a_{t,j}},w)(f_s)_0. \end{aligned}$$

(14)

Lemma 4.1

In the above setting, we have $w(\lambda _{\phi ',0,a_{t,j}})=\lambda _{\phi ',0,a_{t,j}}$ for $w\in W$ such that $w(\Delta {\setminus } \{\alpha \})>0$ if and only if $w=w_0$ or $w=1.$

Proof

Indeed, by (Lemma 4.4 of [23]), (we let $i=a_{t,j}$) the set $\{w\in W:w(\Delta {\setminus } \{\alpha _i\})>0\}$ can be described as $\cup _{0\le j\le i}W_j$ where each $W_j$ consists of elements of the form $p\varepsilon ,$ where $p\in S_n$ and $\varepsilon =(\varepsilon _1,\varepsilon _2,\ldots \varepsilon _n)\in \{\pm 1\}^{n},$ such that the action of $p$ and $\varepsilon$ can be described as (the last row is a description of the $\varepsilon$-action)

We write down $\lambda _{\phi ',0,a_{t,j}}$ as follows:

$$\begin{aligned} \lambda _{\phi ',0,a_{t,j}}=\mu _t\nu ^{-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t\nu ^{\frac{i-1}{2}}\otimes \lambda _{i+1}\nu ^{s_{i+1}}\otimes \cdots \otimes \lambda _n\nu ^{s_n}. \end{aligned}$$

Assume that $w(\lambda _{\phi ',0,a_{t,j}})=\lambda _{\phi ',0,a_{t,j}}$ and $w=p\varepsilon \in W_j$ describe above. Because of our construction, if for some $\lambda _k,\,k\in \{i+1,\ldots , n\}$ we have $\lambda _k=\mu _t,$ then $|s_k|<\frac{i-1}{2}.$ We observe how to get $\mu _t\nu ^{\frac{i-1}{2}}$, i.e., the factor on the $i$th place. Since it cannot come from the factors on the places from $i+1,\ldots n$ (as we just observed), it must come from either factor on the first place ($\mu _t\nu ^{-\frac{i-1}{2}}$) either from the factor on the $i$th place ($\mu _t\nu ^{\frac{i-1}{2}}$). Assume it comes from the factor on the first place. This means $p(1)=i$ and $\varepsilon _1=-1.$ This means $j=0,\,p(1)=i,$ and $p$ is decreasing on $\{1,2,\ldots ,i\}.$ This means $p(2)=i-1,\ldots ,p(i)=1,$ and $p|_{\{i+1,\ldots ,n\}}=id,$ so that $w=p\varepsilon =w_0.$ On the other hand, if it comes from the factor on the $i$th place ($\mu _t\nu ^{\frac{i-1}{2}}$), this means $p(i)=i$ and $\varepsilon _i=1.$ This means $j=i$ and $p(1)=1,\ldots ,p(i)=i,$ also $p|_{\{i+1,\ldots ,n\}}=id,$ so that $w=p\varepsilon =1.$ $\square$

From this Lemma follows that in (14) the second line cannot cancel the expression in the first line for $s=0.$

We have the following important ingredient.

Lemma 4.2

The following holds:

$$\begin{aligned} \lim _{s\rightarrow 0}r(\lambda _{\phi ',s,a_{t,j}},w_0)^{-1}=1. \end{aligned}$$

Note here that if $\mu _t=1,$ then $a_{t,j}>1$ because of the description of the basis of our inductive procedure.

Proof

(of the Lemma) Note that, according to the notation in Lemma 4.1, the element $w_0$ belongs to the set $W_0.$ Now, according to [23] and Corollary 1-3 of [18], we have explicit description of roots $\alpha \in \Sigma ^{+}$ such that $w_0(\alpha )<0,$ and these roots play a role in the definition of the normalizing factor $r(\lambda _{\phi ',s,a_{t,j},v},w_0).$ Let $i=a_{t,j}.$ Then the set of those $\alpha \in \Sigma ^{+}$ such that $w_0(\alpha )<0$ can be divided in several sets:

• (i) $\{2e_k: 1\le k\le i\},$

• (ii) $\{e_k+e_l:1\le k<l\le i\},$

• (iii) $\{e_k-e_l: 1\le k\le i, i+1\le l\le n\},$

• (iv) $\{e_k+e_l:1\le k\le i, i+1\le l\le n, p(l)>p(k)\}.$

Now we study the expression

$$\begin{aligned} \prod _{\alpha \in \sum ^{+}, w_0(\alpha )<0} \frac{L(1,\lambda _{\phi ',s,a_{t,j}}\circ \check{\alpha })\varepsilon (0,\lambda (\underline{s}) \circ \check{\alpha })}{L(0,\lambda (\underline{s})\circ \check{\alpha })} \end{aligned}$$

for $\alpha$ from the each of the above four groups of roots. The contribution from the first group of roots is given by

$$\begin{aligned} \prod _{k=1}^i\frac{L\left( -\frac{i-1}{2}+s+k,\mu _t\right) \epsilon \left( -\frac{i-1}{2}+s+k-1,\mu _t\right) }{L\left( -\frac{i-1}{2}+s+k-1,\mu _t\right) }. \end{aligned}$$

(15)

The contribution from the second group of roots is given by

$$\begin{aligned} \prod _{1\le k<l\le i}\frac{L(2s-i+k+l,1)\epsilon (2s-i+k+l-1,1)}{L(2s-i+k+l-1,1)}. \end{aligned}$$

(16)

Note that in the fourth set of roots, the condition $p(l)>p(k)$ is automatically satisfied, due to the description of the permutation $p$ in $w_0=p\epsilon$ given in the proof of 4.1. So the contribution of the third and the fourth group of roots is given by

$$\begin{aligned}&\prod _{1\le k\le i, i+1\le l\le n}\frac{L\left( s-\frac{i-1}{2}+k-l',\mu _t\chi _l\right) L\left( s-\frac{i-1}{2}+k+l',\mu _t\chi _l\right) }{L\left( s-\frac{i-1}{2}+k-l'-1,\mu _t\chi _l\right) L\left( s-\frac{i-1}{2}+k+l'-1,\mu _t\chi _l\right) }\nonumber \\&\quad \cdot \epsilon \left( s-\frac{i-1}{2}+k-l'-1,\mu _t\chi _l\right) \epsilon \left( s-\frac{i-1}{2}+k+l'-1,\mu _t\chi _l\right) \end{aligned}$$

(17)

Here $\chi _l\nu ^{l'}$ denotes a character appearing as a factor in $\lambda _{\phi ',s,a_{t,j}}$ at the $l$th place ($\chi _l$ is quadratic). We now use a functional equation satisfied by the $L$-functions: for a quadratic Grossencharacter $\chi$ we have

$$\begin{aligned} L(s,\chi )=\epsilon (s,\chi )L(1-s,\chi ). \end{aligned}$$

Now we apply this on the denominators in the expression (15) and then introduce the a change of variables $k\mapsto i+1-k$ to obtain

$$\begin{aligned} \prod _{k=1}^i\frac{L\left( s-\frac{i-1}{2}+k,\mu _t\right) }{L\left( -s-\frac{i-1}{2}+k,\mu _t\right) }. \end{aligned}$$

If $\mu _t\ne 1\, L(s,\mu _t)$ is holomorphic in $s,$ so that each of the above factors tends to $1$ when $s\rightarrow 0.$ If $\mu _t=1,$ we have the following simple fact

$$\begin{aligned} \lim _{s\rightarrow 0}\frac{L(-s+t,1)}{L(s+t,1)}={\left\{ \begin{array}{ll}1;\text { if }t\notin \{0,1\}\\ -1; \text { if }t\in \{0,1\}. \end{array}\right. } \end{aligned}$$

This means that exactly for $k=\frac{i-1}{2}$ and for $k=\frac{i+1}{2}$ the factors are equal to $-1$ when $s\rightarrow 0$ so that overall product consisting from the contributions of the first set of roots equals $1$ (unless $i=a_{t,j}=1$ and $\mu _t=1$ so that whole product consists only of one factor, but this cannot happen).

After applying the functional equation on the denominators of the contribution (16) and then applying change of variables $(k,l)\mapsto (i+1-l,i+1-k)$ we obtain

$$\begin{aligned} \prod _{1\le k<l\le i}\frac{L(2s-i+k+l,1)}{L(-2s-i+k+l,1)}. \end{aligned}$$

This means that in the cases when $k+l=i$ and $k+l=i+1$ these factors tend to $-1$ when $s\rightarrow 0.$ There are exactly $2\frac{i-1}{2}=i-1$ (even number!) of these instances, so the overall product tends to $1.$ If $i=1$ there is no second product.

After applying the functional equation on the denominators in the expression (17) and applying the change of variables $k\mapsto i+1-k,$ and then grouping together numerators and denominators in a nice way, we obtain

$$\begin{aligned}&\prod _{i+1\le l\le n}\prod _{1\le k\le i}\frac{L\left( s+2+\frac{i-1}{2}-k-l',\mu _t\chi _l\right) }{L\left( -s+2+\frac{i-1}{2}-k-l',\mu _t\chi _l\right) }\cdot \nonumber \\&\prod _{i+1\le l\le n}\prod _{1\le k\le i}\frac{L\left( s+2+\frac{i-1}{2}-k+l',\mu _t\chi _l\right) }{L\left( -s+2+\frac{i-1}{2}-k+l',\mu _t\chi _l\right) }. \end{aligned}$$

(18)

We have to check what happens if $\chi _l=\mu _t.$ The number of times when $l'$ is such that $2+\frac{i-1}{2}-k-l'=1$ for any $k\in \{1,2,\ldots ,i\}$ is equal to the number of $l'$ such that $2+\frac{i-1}{2}-k+l'=1$ any $k\in \{1,2,\ldots ,i\}.$ Analogously we cancel of the $-1$’s in the numerators, keeping in mind that $|l'|<\frac{i-1}{2}$ (since we study the exponents corresponding to $\mu _t$). $\square$

Now we have that (so $|S|$ is even)

$$\begin{aligned}&\lim _{s\rightarrow 0}(f_s)_0 +r(\lambda _{\phi ',s,a_{t,j}},w_0)^{-1}\otimes _{v\notin S}f_{w_0(s),v}\otimes _{v\in S} N(\mu _{t,GL(a_{t,j},k_v)}\nonumber \\&\quad \otimes \sigma _v(\phi '),w_0)f_{s,v}= 2(f_0)_0\ne 0, \end{aligned}$$

since, $|S|$ is even and for $v\in S$ we have $N(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma _v(\phi '),w_0)f_{0,v}=-f_{0,v},$ so for such choice of $\pi$ we have that intertwining (9) is non-trivial on $\pi$ (since after calculation of the constant term we still get a non-zero contribution) and the Claim is proved.

4.2 Step 2

Proposition 4.3

Assume that $\pi$ is an irreducible subrepresentation of (8) such that the intertwining (9) does not send it to zero (e.g., one for which we take $|S|$ to be even). Let $E(\pi )$ be the image of $\pi$ under this intertwining (so that $E(\pi )\subset A(Sp_{2n'+2a_{t,j}}(k)){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))$). Let $E_0(\pi )$ be a constant term along $B_{n'+a_{t,j}}.$ Now we construct degenerate Eisenstein series $f_s\mapsto E(f_s,g)=\sum _{\gamma \in P(k){\setminus } Sp_{2n}(k)}f_s(\gamma g)$ attached to the global induced representation $\mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\mu _t|det|^s\otimes E_0(\pi )),$ where $P$ is the standard parabolic subgroup with Levi factor isomorphic to $GL(\frac{a_{t,j+1}-a_{t,j}}{2})\times Sp_{2n'+2a_{t,j}}$. Then, map

$$\begin{aligned} \mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\mu _t|det|^{\frac{a_{t,j}+a_{t,j+1}}{4}}\otimes E_0(\pi ))\rightarrow A(Sp_{2n}(k){\setminus } Sp_{2n}(\mathbb {A})) \end{aligned}$$

given by

$$\begin{aligned} f_{{\frac{a_{t,j}+a_{t,j+1}}{4}}}\rightarrow \left( s-{\frac{a_{t,j}+a_{t,j+1}}{4}}\right) ^2E(f_s,\cdot )|_{s={\frac{a_{t,j}+a_{t,j+1}}{4}}} \end{aligned}$$

is well-defined and non-trivial, and its image $E(\pi ,\mu _t,a_{t,j+1})$ is contained in the space of square-integrable automorphic forms. Every irreducible subrepresentation of $E(\pi ,\mu _t,a_{t,j+1})$ (which is semi-simple representation) is thus automorphic square-integrable, and has a global Arthur parameter equal to $\phi .$

Proof

We denote

$$\begin{aligned} \lambda _s=\mu _t|\cdot |^{s-\frac{\frac{a_{t,j+1}-a_{t,j}}{2}-1}{2}}\otimes \cdots \otimes \mu _t|\cdot |^{s+\frac{\frac{a_{t,j+1}-a_{t,j}}{2}-1}{2}}\otimes \lambda _{\phi ',0,a_{t,j}}, \end{aligned}$$

and $\alpha$ denote a simple root such that the standard parabolic subgroup $P$ of $Sp_{2n}$ from the statement of Proposition corresponds to set of roots $\Delta {\setminus } \{\alpha \}.$ Let $S'$ be a finite set of places containing the archimedean ones. Let $f_s=\otimes _v (f_s)_v$ be a factorizable function in the space of $\mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\mu _t|det|^s\otimes E_0(\pi ))$ such that for $v\notin S',$ the function $f_s$ is $K_v$-invariant. We calculate the constant term of the degenerate Eisenstein series $f_s\mapsto E(f_s,\cdot );$ similarly as in the first step of the construction, but now, since we immediately realized $\pi$ in the space of it’s constant terms (cf. [17, section 2]) and embed $\mu _t|det|^s\hookrightarrow \mu _t|\cdot |^{s-\frac{\frac{a_{t,j+1}-a_{t,j}}{2}-1}{2}}\otimes \cdots \otimes \mu _t|\cdot |^{s+\frac{\frac{a_{t,j+1}-a_{t,j}}{2}-1}{2}}$ (also by taking an appropriate constant term), we get

$$\begin{aligned} E_0(s,f_s)(g)&= \sum _{w\in W_n,w(\Delta {\setminus } \{\alpha \})>0}M(\lambda _s,w)f_s\\&= M(\lambda _s,w_0)f_s+\sum _{w\in W_n,w(\Delta {\setminus } \{\alpha \})>0,w\ne w_0}M(\lambda _s,w)f_s\\&= r(\lambda _s,w_0)^{-1}(\otimes _{v\notin S'} f_{w_0(\lambda _s,v)})\otimes _{v\in S'}N(s,\mu _{t,v}\otimes (E_0(\pi ))_v,w_0)f_{s,v})\\&\quad +\sum _{w\in W_n,w(\Delta {\setminus } \{\alpha \})>0,w\ne w_0}r(\lambda _s,w)^{-1}(\otimes _{v\notin S'} f_{w(\lambda _s,v)})\\&\otimes _{v\in S'}N(s,\mu _{t,v} \otimes E_0(\pi ))_v,w)f_{s,v}). \end{aligned}$$

Here we use the analog of the normalization for the operators $N(s,\mu _{t,v}\otimes (E_0(\pi ))_v,w_0)$ described in the previous proposition, and use the fact that (un)normalized operators $M(\lambda _s,w)$ and $M(s,\mu _{t}\otimes E_0(\pi ))$ (and, consequently, $N(\lambda _s,w)$ and $N(s,\mu _{t}\otimes (E_0(\pi )),w)$) agree on $\mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\mu _t|det|^s\otimes E_0(\pi ))$ which is, by taking a constant term, realized as a subspace of $\mathrm{Ind}_{B_n(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\lambda _s)$ (cf. section 3 of [18]).

We are interested in the situation for $s=s_0=\frac{a_{t,j}+a_{t,j+1}}{4}.$ We have an important lemma.

Lemma 4.4

In the above situation, the order of the pole at $s=\frac{a_{t,j}+a_{t,j+1}}{4}$ of $r(\lambda _s,w_0)^{-1}$ is two, and of $r(\lambda _s,w)^{-1}$ for $w\ne w_0$ is at most one. Note that either $a_{t,j}>1$ or $\mu _t\ne 1.$

Proof

We first calculate the order of pole for $s=s_0$ of $r(\lambda _s,w_0)^{-1}.$ Again,we use the similar notation as in the proof of Lemma 4.2. Let now $i= \frac{a_{t,j+1}-a_{t,j}}{2}.$ We return to the contributions (15)–(17). We now want to examine possible poles and zeros of these expressions for $s_0=\frac{a_{t,j}+a_{t,j+1}}{4}.$ It is easy to see that the enumerator of the expression (15) does not have any poles or zeros, and the denominator has no zeros and a simple pole if $a_{t,j}=1$ and $\mu _t=1,$ but this is impossible here. As for the expression (16), we see that, since both numerator and denominator express values for $L$-function $L(z,1)$ for (real) $z>2$ there are no zeros or poles. In (17) there are no poles in the numerator if $\chi _l=\mu _t,$ and there is a pole of second order in the denominator (if $\chi _l=\mu _t,$ and that happens exactly for the factors with $l'=\nu ^{-\frac{a_{t,j}-1}{2}}$ and $l'=\nu ^{\frac{a_{t,j}-1}{2}},$ and these factors do appear in $\lambda _{\phi ',0,a_{t,j}},$ as we have seen in the previous step of the construction).

If $l$ is such that $\chi _l\ne \mu _t,$ the corresponding $L$-functions are entire, but there are also no zeros appearing since $L(*, \mu _t\chi _l)$ is evaluated at the rational integers. This means that $r(\lambda _s,w_0)$ has a zero of order $2$ at $s=s_0$ if $a_{t,j}\ne 1$ and of order $3$ if $a_{t,j}=1$ and $\mu _t=1$ and the claim about poles of $r(\lambda _s,w_0)^{-1}$ follows.

Now we deal with $w\ne w_0.$ Let $w=p\epsilon \in W_j$ (what this means is described in the proof of Lemma 4.1). We need to describe all $\alpha \in \Sigma ^+$ such that $w(\alpha )<0$ to calculate the normalizing factor $r(\lambda _s,w).$ The set of these $\alpha$’s consists of six sets

• 1. $S_1=\{2e_k:j+1\le k\le i\},$

• 2. $S_2=\{e_k+e_l:j+1\le k<l\le i\},$

• 3. $S_3=\{e_k-e_l:j+1\le k\le i,\;i+1\le l\le n\},$

• 4. $S_4=\{e_k+e_l:1\le k\le j,j+1\le l\le i,\;p(l)<p(k)\},$

• 5. $S_5=\{e_k-e_l:1\le k\le j,i+1\le l\le n,\;p(l)<p(k)\},$

• 6. $S_6=\{e_k+e_l:j+1\le k\le i,i+1\le l\le n,p(l)>p(k)\}.$

We thus have

$$\begin{aligned} r(\lambda _s,w)=\prod _{t=1}^6\prod _{\alpha \in S_t}\frac{L(1,\lambda _s\circ \check{\alpha })\varepsilon (0,\lambda _s\circ \check{\alpha })}{L(0,\lambda _s\circ \check{\alpha })}. \end{aligned}$$

Now, reasoning analogously as in the case of $w_0,$ we see that the contribution (in $L$-functions) to $r(\lambda _{s_0},w)$ is holomorphic (and non-zero) for $\alpha \in S_1\cup S_2\cup S_4.$ There is a pole of the first order of the denominator for $\alpha \in S_3$ if $k=1,$ so that $j=0,$ and for $\alpha \in S_5$ if $k=1$ so that $j\ge 1.$ Analogously, $\alpha \in S_6$ contributes with the pole of the first order in the denominator if $k=1,$ i.e., $j=0.$ (Again, we have a separate case if $a_{t,j}=1$ and $\mu _t=1,$ in that case, $\alpha \in S_1$ also contributes if $k=1,$ so $j=0$). Assume now that $a_{t,j}>1$ or $\mu _t\ne 1.$ Then if $r(\lambda _{s},w)^{-1}$ has a pole of second order for $s=s_0,$ we have $j=0,$ and $p(i+1)>p(1)$ (this comes from the condition on $\alpha \in S_6$). Now, from the scheme in the proof of Lemma 4.1, we see that $w=w_0.$ $\square$

We denote $r(w_0)=\lim _{s\rightarrow s_0}(s-s_0)^2r(\lambda _s,w_0)^{-1}\ne 0.$ Lemma 4.4 insures that

$$\begin{aligned}&\lim _{s\rightarrow s_0}(s-\frac{a_{t,j}+a_{t,j+1}}{4})^2E_0(s,f_s)(g)\nonumber \\&\quad =r(w_0)(\otimes _{v\notin S'} f_{w_0(\lambda _{s_0},v)})\otimes _{v\in S'}N(s_0,\mu _{t,v}\otimes (E_0(\pi ))_v,w_0)f_{s_0,v}). \end{aligned}$$

Indeed, the operator $N(s_0,\mu _{t,v}\otimes (E_0(\pi ))_v,w)$ is holomorphic at $s=s_0=\frac{a_{t,j}+a_{t,j+1}}{4}$ for all $w\in W_n,w(\Delta {\setminus } \{\alpha \})>0$ (Lemma 3.5(i) of [17]) so we get the limit above (we just calculate $r(\lambda _s,w)^{-1}$ as $s\rightarrow s_0$). Also, $M(s_0,\mu _{t,v}\otimes (E_0(\pi ))_v,w_0)$ is nonzero for $s=s_0,$ so we get that $\lim _{s\rightarrow s_0}(s-\frac{a_{t,j}+a_{t,j+1}}{4})^2E_0(s,f_s)(g)$ is non-zero, specifically $\lim _{s\rightarrow s_0}(s-\frac{a_{t,j}+a_{t,j+1}}{4})^2E(f_s,g)=E(\pi ,\mu _t,a_{t,j+1})$ is non-zero, and $E(\pi ,\mu _t,a_{t,j+1})$ belongs to the space of the square-integrable automorphic forms on $Sp_{2n}(k){\setminus } Sp_{2n}(\mathbb {A})$ (as we can see from the calculation of it’s constant term). We see that the constant term of $E(\pi ,\mu _t,a_{t,j+1})$ (which is isomorphic to this representation) is generated, for all $v\notin S',$ by the spherical vector $f_{w_0(\lambda _{s_0},v)}$ inside $\mathrm{Ind}_{P(k_v)}^{Sp_{2n}(k_v)}(\mu _{t,v}|det|_v^s\otimes E_0(\pi )_v)\hookrightarrow \mathrm{Ind}_{B_n(k_v)}^{Sp_{2n}(k_v)}(w_0(\lambda _{s_0,v})).$

Note that $w_0(\lambda _{s_0,v})=\lambda (\phi )_v,$ the (local component) of a character of the maximal torus, attached to the Arthur parameter $\phi .$

Lemma 4.5

For all $v\notin S'$ the subspace in $\mathrm{Ind}_{B_n(k_v)}^{Sp_{2n}(k_v)}(w_0(\lambda _{s_0,v}))$ generated by the spherical vector is an irreducible (spherical) subrepresentation $\sigma _v$ of $\mathrm{Ind}_{B_n(k_v)}^{Sp_{2n}(k_v)}(w_0(\lambda _{s_0,v})).$

This lemma follows from Proposition 1.2 and it guarantees that every irreducible subrepresentation of $E(\pi ,\mu _t,a_{t,j+1})$ is attached to the Arthur parameter $\phi .$ $\square$

5 Explicit construction II: representations with general support

To be able to execute the same steps for general case of automorphic unipotent representations (as we have done in the previous section in the case of such representation with some constant term non-vanishing along the Borel subgroup) we need some additional results. One of these results actually states that any automorphic representation, occurring as a subrepresentation in the space of automorphic forms (so especially, a representation occurring in the discrete part of square-integrable spectrum) can be embedded (as an abstract representation) in the representation induced from a (essentially) cuspidal representation of a Levi subgroup. This result is proved in [10]. For our purposes, we need an explicit realization of the space of the induced representation, and we actually have that same realization in [10]. This result is straightforward application of Langlands’ method in his Corvallis lecture combined with various decompositions of the spaces of automorphic forms which can be found in the second and the fourth chapter of [15], and we briefly explain it.

Assume that we have an irreducible representation realized in the space of automorphic forms $Sp_{2n'}(k){\setminus } Sp_{2n'}(\mathbb {A})$ (in this setting, it is important to be aware of the explicit realizations of the global representations in certain spaces of automorphic forms). Assume that this representation, say $(\Pi ,V),$ is concentrated on the (standard) parabolic subgroup $P_{\theta _0},$ (parabolic subgroup which corresponds to the subset $\theta _0$ of the set of simple roots of $Sp_{2n'}$ with respect to the maximal diagonal torus (and upper triangular Borel subgroup)), and assume that $\theta _0$ is minimal with this property. This means that the constant term of $(\Pi ,V)$ along $P_{\theta _0}$ does not vanish. We show in [10] that there is a character $\xi$ of $Z_{M_{\theta _0}}(\mathbb {A})$ such that (maybe after left translation by an element of $Z_{M_{\theta _0}}(\mathbb {A})$), the space of constant terms of $(\Pi ,V),$ denoted $V_0\ne 0,$ belongs to the space of automorphic forms $A_0(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A}))_\xi ,$ (for the notation and the definitions of these spaces of automorphic forms we refer to [15], I.2.17, and for the constant term I.2.6). Then we show that there is an automorphic representation $\pi _0$ of $M_{\theta _0}(\mathbb {A})$ (cuspidal, because of the minimality of the set $\theta _0$) such that $V_0$ belongs to $A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A}))_{\pi _0}$ [15, II.1.1.]. We recall that $A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A}))_{\pi _0}$ is defined as

$$\begin{aligned} A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A}))_{\pi _0}&= \{\phi \in A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A})):\\&\forall k \in K,\phi _k\in A(M_{\theta _0}(k){\setminus } M_{\theta _0}(\mathbb {A}))_{\pi _0}\} \end{aligned}$$

where $\phi _k(m)=\delta _{P_{\theta _0}}^{-\frac{1}{2}}\phi (mk);$ this space can canonically be identified with

$$\begin{aligned} \mathrm{ind}_{M_{\theta _0}(\mathbb {A})\cap K}^KA(M_{\theta _0}(k){\setminus } M_{\theta _0}(\mathbb {A}))_{\pi _0}, \end{aligned}$$

hence the statement about embedding in the induced representation (as abstract representations).

So, to conclude, the following holds [10].

Theorem 5.1

Let $(\Pi ,V)$ be a ($(\mathfrak {g}_{\infty },K_{\infty })\times \prod _{v<\infty }Sp_{2n'}(k_v)$)-irreducible subspace of the space of automorphic forms inside $A(Sp_{2n'}(k){\setminus } Sp_{2n'}(\mathbb {A}))$ such that some constant term of the functions from $V$ does not vanish along a parabolic subgroup $P_{\theta _0}$ of $Sp_{2n'};$ assume that $\theta _0$ is minimal (set of simple roots) with this property. Then, there exists an irreducible automorphic representation $\pi _0$ of $M_{\theta _0}(\mathbb {A})$ (appearing in $A_0(M_{\theta _0}(k){\setminus } M(\mathbb {A}))$ such that the space of constant terms of $V$ along $P_{\theta _0},$ denoted by $V_0,$ belongs (up to a left translation by an element from $Z_{M_{\theta _0}})$) to the space $A_0(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A}))_{\pi _0}$ of cuspidal automorphic forms.

Remark

The embedding of the representation $\Pi$ inside the space $A_0(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'}(\mathbb {A}))_{\pi _0}$ given in this theorem is referred to as “given by the constant term.”

We show (Theorem 5.3) that the constant term of certain (degenerate Eisenstein series) related to the representation $\pi$ as above decomposes as a sum of global intertwining operators, similarly to the case of (ordinary) Eisenstein series.

Calculation of this sort were executed also in [18], but only in the case $\theta _0=\emptyset ,$ i.e., when the representation $\Pi$ is supported on the Borel subgroup. To treat this (more general) case, we need some additional results from [15].

So, recall that in the first step of our inductive construction we use (a degenerate) Eisenstein series as an intertwining operator from the space of automorphic forms related to the space from the previous theorem, equivalent to a certain induced representation, to the space of automorphic forms on $Sp_{2n''}(k){\setminus } Sp_{2n''}(\mathbb {A}),$ for certain $n''$ to be defined later. The local components of this induced (unitary) representation will chosen so that the resulting representation has a non-zero Eisenstein series; this would be guaranteed by the fact that the constant term of this Eisenstein series (along the parabolic $P_{\theta _0}$) is non-zero.

5.1 Step 1

Let $\phi '$ be the Arthur parameter build up in the previous step of our inductive procedure, and let $\sigma =\otimes _v\sigma _v$ be an irreducible automorphic representation of $Sp_{2n'}(\mathbb {A})$ with the Arthur parameter $\phi '$ realized in the space of the automorphic forms $A(Sp_{2n'}(k){\setminus } Sp_{2n'}(\mathbb {A})).$ We assume that the basis of our inductive procedure is Arthur parameter $\phi _0$, and that we have an automorphic representation $\sigma _0$ of $Sp_{2n_0}(\mathbb {A})$ with that Arthur parameter, and that $\sigma _0$ is concentrated on the parabolic subgroup $P_{\theta _0}.$ Now, as we add new elements in the Jordan block, and the rank of the symplectic group in question raises, we always make this convention about the notation of a subset $\theta _0$ of the set of simple roots: let say that $\theta _0$ is a subset of the set of simple roots of $Sp_{2n_0}$ such that $M_{\theta _0}\cong GL_{n_1}\times GL_{n_2}\times \cdots \times GL_{n_k}\times Sp_{2m_1}$ (of course, depending on $\theta _0,$ the last $Sp_{m_1}$ does not have to be there). Now, we enlarge the rank of the symplectic group; say that we have $Sp_{2n'}$ now. Now, the simple roots in $Sp_{2n'}$ are rearranged in a such away that $M_{\theta _0}'$ (Levi subgroup in $Sp_{2n'}$ attached to $\theta _0$) is isomorphic to $GL_1\times \cdots \times GL_1\times GL_{n_1}\times \cdots \times GL_{n_k}\times Sp_{m_1},$ where the number of $GL_1'$s at the beginning is exactly $n'-n_0.$ With this convention, we have the following lemma and theorem.

We need next observation about the action of the elements of the Weyl group $W$ of $Sp_{2n}.$

Lemma 5.2

Let $\alpha _i=e_i-e_{i+1}$ be a simple root with $i\le n-n_0$ and $\emptyset \ne \theta _0\subset \Delta (Sp_{2n_0}).$ Let $w\in W$ be such that $w(\Delta {{\setminus }} \{\alpha _i\})>0,$ and $w(\theta _0)=\theta _0.$ Then, $w$ acts as the identity on $\theta _0,$ moreover it acts as a identity on every simple root of $\Delta (Sp_{2n_0})$ “between” the simple roots in $\theta _0.$ As a consequence, if $e_{n-n_0+1}-e_{n-n_0+2}\in \theta _0$ (the first simple root of $Sp_{2n_0}$ in the standard ordering with our convention above about the numeration of roots), then $w$ acts as identity on $\Delta (Sp_{2n_0}).$

Proof

We use the description of those elements of Weyl group for which $w(\Delta {\setminus } \{\alpha _i\})>0$ given in Lemma 4.1. Since $w=p$ (in the notation of this Lemma) is increasing on $\{e_{n-n_0+1},\ldots ,e_n\}$ and $w(\theta _0)=\theta _0,$ it is obvious that $w$ acts on $\theta _0$ as an identity, and also it then must act as a identity on the simple roots in $\Delta (Sp_{2n_0})$ between the roots in $\theta _0.$ $\square$

For this theorem we also define $i=a_{t,j};$ $W$ denotes the Weyl group of $Sp_{2n'+2a_{t,j}}$ attached to our standard choices of maximal torus and Borel subgroup. The standard parabolic subgroups (and their unipotent radicals) of the group $Sp_{2n'}$ are denoted by prime. Calculation in the next theorem is essentially the one from [15, II.1.7], but we had to adapt it, since the inducing data in our case is not cuspidal so we had to use a bit different realizations of global representations.

Theorem 5.3

Let $\sigma (\phi ')$ be an automorphic representation of $Sp_{2n'}(\mathbb {A})$ and $\phi '$ Arthur parameter of $\sigma (\phi '),$ as described above. We assume that $\sigma (\phi ')$ is concentrated on $P_{\theta _0}',$ and let $\pi _0'$ be an automorphic cuspidal representation of $M_{\theta _0}(\mathbb {A})'$ such that $\sigma (\phi ')$ is realized as a subspace in $A(N_{\theta _0}'(\mathbb {A})M_{\theta _0}'(k){\setminus } Sp_{2n'}(\mathbb {A}))_{\pi _0'}$ (cf. Theorem 5.1). We consider the global induced representation

$$\begin{aligned} \pi _s=\mathrm{Ind}_{P_{\Delta {\setminus }\{\alpha _i\}}(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi ')). \end{aligned}$$

(19)

realized in the space $A(U_{\Delta {\setminus } \alpha _i}(\mathbb {A})M_{\Delta {\setminus } \alpha _i}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))_{|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi ')}$(keeping in mind our realization of $\sigma (\phi ')$). For $f_s$ from this space, we define Eisenstein series

$$\begin{aligned} E(s,f_s)(g)=\sum _{\gamma \in P_{\Delta {\setminus } \{\alpha _i\}}(k){\setminus } Sp_{2n'+2a_{t,j}}(k)}f_s(\gamma g), \end{aligned}$$

cf. [15, II.1.5]. The constant term of $E(s,f_s)$ along $P_{\theta _0}$ (a standard parabolic of $Sp_{2n'+2a_{t,j}}(\mathbb {A})$) is given by

$$\begin{aligned} E_{P_{\theta _0}}(s,f_s)(g)=\sum _{w\in W,w(\Delta {\setminus } \alpha _i)>0,w(\theta _0)=\theta _0}M(w,\pi ,s)(f_s)(g) \end{aligned}$$

where

$$\begin{aligned}&M(w,\pi ,s):A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))_{\mu _t \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{s+\frac{i-1}{2}}\otimes \pi _0'} \nonumber \\&\quad \rightarrow A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))_{w(\mu _t \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{s+\frac{i-1}{2}}\otimes \pi _0')} \end{aligned}$$

is an intertwining operator.

Proof

Let $f_s\in A(U_{\Delta {\setminus } \alpha _i}(\mathbb {A})M_{\Delta {\setminus } \alpha _i}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))_{|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi ')}.$ We have the following disjoint decomposition [7] (since $Sp_{2n'+2a_{t,j}}$ is split over $k,$ this decomposition works over $k$ and over $\mathbb {A}$):

$$\begin{aligned} Sp_{2n'+2a_{t,j}}=\bigcup _{\begin{array}{c} w\in W,w^{-1}(\theta _0)>0,\\ w(\Delta {\setminus } \{\alpha _i\})>0 \end{array}}P_{\Delta {\setminus } \{\alpha _i\}}w^{-1}P_{\theta _0}, \end{aligned}$$

where $P_*$ and $N_*$ denote a standard parabolic subgroup and unipotent radical of a standard parabolic subgroup attached to a subset $*$ of the set of simple roots. For such $w\in W,$ easily follows that

$$\begin{aligned} P_{\Delta {\setminus } \{\alpha _i\}}w^{-1}P_{\theta _0}=P_{\Delta {\setminus } \{\alpha _i\}}w^{-1}(N_{\theta _0}\cap w\overline{N_{\emptyset }}w^{-1})(M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1}), \end{aligned}$$

where

$$\begin{aligned}&P_{\Delta {\setminus } \{\alpha _i\}}\times (N_{\theta _0}\cap w\overline{N_{\emptyset }}w^{-1})\times (M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1}) \nonumber \\&\quad \rightarrow P_{\Delta {\setminus } \{\alpha _i\}}w^{-1}(N_{\theta _0}\cap w\overline{N_{\emptyset }}w^{-1})(M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1}) \end{aligned}$$

given by $(x,y,z)\mapsto xw^{-1}yz$ is an isomorphism of varieties (cf, e.g., [5, 14.12]). Now we calculate the constant term of the Eisenstein series in question:

$$\begin{aligned} E_{P_{\theta _0}}(s,f_s)(g)\!&= \!\int _{N_{\theta _0}(k){\setminus } N_{\theta _0}(\mathbb {A})}\sum _{\gamma \in P_{\Delta {\setminus } \{\alpha _i}\}(k){\setminus } Sp_{2n'+2a_{t,j}}(k)}f_s(\gamma ug)\mathrm{d}u\nonumber \\ \!\!&= \!\!\int _{N_{\theta _0}(k){\setminus } N_{\theta _0}(\mathbb {A})}\left( \sum _{\begin{array}{c} w\in W,w^{-1}(\theta _0)>0,\!\\ w(\Delta {\setminus } \{\alpha _i\})>0 \end{array}}\sum _{\begin{array}{c} \gamma \in (N_{\theta _0}\cap w\overline{N_{\emptyset }}w^{\!-1})(k)\\ (M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1})(k) \end{array}}f_s(w^{-1}\gamma ug)\right) \!\mathrm{d}u. \end{aligned}$$

We can exchange the sum and the integral because of the convergence properties of the series, $N_{\theta _0}(k){\setminus } N_{\theta _0(\mathbb {A})}$ being compact [15, II.1.5.]. Since $N_{\theta _0}(k)=(N_{\theta _0}(k) \cap wN_{\emptyset }w^{-1})(N_{\theta _0}(k) \cap w\overline{N_{\emptyset }}w^{-1}),$ we can write down the last line as

$$\begin{aligned}&\sum _{\begin{array}{c} w\in W,w^{-1}(\theta _0)>0,\\ w(\Delta {\setminus } \{\alpha _i\})>0 \end{array}}\sum _{m\in (M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1})(k)}\int _{N_{\theta _0}(k)\cap wN_{\emptyset }w^{-1}(k){\setminus } N_{\theta _0}(\mathbb {A})}f_s(w^{-1}umg)\mathrm{d}u\nonumber \\&\quad =\sum _{\begin{array}{c} w\in W,w^{-1}(\theta _0)>0,\\ w(\Delta {\setminus } \{\alpha _i\})>0 \end{array}}\sum _{m\in (M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1})(k)}\int _{N_{\theta _0}(k)\cap wN_{\emptyset }w^{-1}(k){\setminus } N_{\theta _0}(\mathbb {A})\cap wN_{\emptyset }w^{-1}(\mathbb {A})}\nonumber \\&\quad \int _{N_{\theta _0}(\mathbb {A})\cap wN_{\emptyset }w^{-1}(\mathbb {A}){\setminus } N_{\theta _0}(\mathbb {A})}f_s(w^{-1}u_1umg)\mathrm{d}u_1\mathrm{d}u. \end{aligned}$$

By the change of the variable $u_1\mapsto w^{-1}u_1w$ the first integral is transformed into integral over $(w^{-1}N_{\theta _0}w\cap N_{\emptyset })(k){\setminus } (w^{-1}N_{\theta _0}w\cap N_{\emptyset })(\mathbb {A}),$ and then we can interchange the order of integration. We then decompose the set of integration of the (now) second integral:

$$\begin{aligned} w^{-1}N_{\theta _0}w\cap N_{\emptyset }=(w^{-1}N_{\theta _0}w \cap M_{\Delta {\setminus } \{\alpha _i\}})(w^{-1}N_{\theta _0}w\cap N_{\Delta {\setminus } \{\alpha _i\}}). \end{aligned}$$

Since $f_s\in A(N_{\Delta {\setminus } \{\alpha _i\}}(\mathbb {A})M_{\Delta {\setminus } \{\alpha _i\}}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A})),$ the integration over $(w^{-1}N_{\theta _0}w\cap N_{\Delta {\setminus } \{\alpha _i\}})$ is irrelevant, so that we have this expression for $E_{P_{\theta _0}}(f_s,g)$

$$\begin{aligned} \sum _{\begin{array}{c} w\in W,w^{-1}(\theta _0)>0,\\ w(\Delta {\setminus } \{\alpha _i\})>0 \end{array}}\sum _{m\in (M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1})(k)}\int \int f_s(u_2w^{-1}umg)\mathrm{d}u_2\mathrm{d}u \end{aligned}$$

(20)

where the first integral is over $N_{\theta _0}(\mathbb {A})\cap wN_{\emptyset }w^{-1}(\mathbb {A}){\setminus } N_{\theta _0}(\mathbb {A})$ and the second over $w^{-1}N_{\theta _0}w \cap M_{\Delta {\setminus } \{\alpha _i\}}(k){\setminus } w^{-1}N_{\theta _0}w \cap M_{\Delta {\setminus } \{\alpha _i\}}(\mathbb {A}).$ Now, since $M_{\Delta {\setminus } \{\alpha _i\}}(\mathbb {A})\cong GL_i(\mathbb {A})\times Sp_{2n'}(\mathbb {A}),$ we can see that for $u_2\in GL_i\cap w^{-1}N_{\theta _0}w$ we have $f_s(u_2*)=f_s(*),$ because of the domain of the definition of $f_s.$ This means that in the expression for $E_{P_{\theta _0}}(s,f_s)(g)$ (20) the second integral is over $(w^{-1}N_{\theta _0}w \cap Sp_{2n'})(k){\setminus } (w^{-1}N_{\theta _0}w \cap Sp_{2n'})(\mathbb {A}).$ Of course, we constantly use our normalization of Haar measures on the unipotent radicals.

It is easy to see that $w^{-1}N_{\theta _0}w \cap Sp_{2n'}=w^{-1}N_{\theta _0}w \cap N_{\emptyset }'$ is a unipotent radical of standard parabolic subgroup of $Sp_{2n'}.$ It is not difficult to show that $(N_{\emptyset }'\cap w^{-1}N_{\theta _0}w)N_{\theta _0}'$ is a unipotent subgroup which is the unipotent radical of a standard parabolic subgroup of $Sp_{2n'}$ attached to the set of roots $w^{-1}\theta _0\cap \theta _0\subset \Delta {'}.$ Because we normalized Haar measures on the unipotent radicals as we did, and in our realization $f_s$ is left invariant under $N'_{\theta _0}(\mathbb {A}),$ the inner integral in (20) can be taken to be over $(N_{\emptyset }'\cap w^{-1}N_{\theta _0}w)N_{\theta _0}'.$ Now, if $w^{-1}\theta _0\cap \theta _0\subsetneq \theta _0$ because of the cuspidality of $\pi _0,$ this integral is $0.$ So, in order for this integral to be non-zero, we have $w(\theta _0)=\theta _0.$ This forces $M_{\theta _0}\cap w\overline{N_{\Delta {\setminus } \{\alpha _i\}}}w^{-1}=\{e\},$ and (20) becomes

$$\begin{aligned} \sum _{\begin{array}{c} w\in W,w(\theta _0)=\theta _0,\\ w(\Delta {\setminus } \{\alpha _i\})>0 \end{array}}\int _{N_{\theta _0}(\mathbb {A})\cap wN_{\theta _0}w^{-1}(\mathbb {A}) {\setminus } N_{\theta _0}(\mathbb {A})} f_s(w^{-1}ug)\mathrm{d}u. \end{aligned}$$

In the previous expression we recognize the intertwining operator

$$\begin{aligned}&M(w,\pi ,s):A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))_{\mu _t \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{s+\frac{i-1}{2}}\otimes \pi _0'} \nonumber \\&\quad \rightarrow A(U_{\theta _0}(\mathbb {A})M_{\theta _0}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A}))_{w(\mu _t \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{s+\frac{i-1}{2}}\otimes \pi _0')}, \end{aligned}$$

as claimed. For the convergence issues and the meromorphic continuation of the (constant term of) the Eisenstein series and intertwining operators, we refer to [15], the second and the fourth chapter. By a result of Langlands (e.g., [2, Theorem 7.2]), this meromorphic continuation of the Eisenstein series is holomorphic for $s=0$ (the realization of $\sigma (\phi ')$ in our case is a bit different than in the cited theorem, but since this realization is given by taking the constant term, the holomorphy argument is the same). $\square$

We continue with the assumptions from the previous theorem.

Proposition 5.4

For $w \in W, w({\theta }_0)=\theta _0, \, w(\Delta {\setminus } \{\alpha _i\})>0$

$$\begin{aligned} w(\mu _t \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{s+\frac{i-1}{2}}\otimes \pi _0')=\mu _t \nu ^{-s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{-s+\frac{i-1}{2}}\otimes \pi _0' \end{aligned}$$

for $s=0$ if and only if $w=1$ or $w=w_0$ (the longest element in $W$ modulo the longest one in $M_{\theta _0}$).

Proof

Assume that

$$\begin{aligned} w(\mu _t \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{s+\frac{i-1}{2}}\otimes \pi _0')=\mu _t \nu ^{-s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _t \nu ^{-s+\frac{i-1}{2}}\otimes \pi _0'. \end{aligned}$$

We fix $s=0$ and analyze how we can get factor $\mu _t\nu ^{\frac{i-1}{2}}.$ Our inductive procedure says that $\pi _0'\cong \lambda _1\nu ^{s_1}\otimes \cdots \otimes \lambda _k\nu ^{s_k}\otimes \pi _0'',$ where $\lambda _1\nu ^{s_1},\ldots \lambda _k\nu ^{s_k}$ are Grossencharacters obtained by adding elements in the Jordan blocks, and $\pi _0''$ is a cuspidal representation of the appropriate Levi subgroup of $Sp_{2n_0},$ associated to the representation $\sigma (\phi _0),$ as explained in Theorem 5.1. Now, similarly as in Lemma 4.1, we see that $\mu _t\nu ^{\frac{i-1}{2}}$ (the $i$th) factor cannot be obtained from $\lambda _1\nu ^{s_1},\ldots \lambda _k\nu ^{s_k}.$ Assume now that $w(\chi _j\nu ^{s_j})=\mu _t\nu ^{\frac{i-1}{2}},$ where $\chi _j\nu ^{s_j}$ is a factor of $\pi _0''$ ($s_j\in \mathbb {R}$). Now we have two cases. If $\theta _0\subset \Delta (Sp_{2n_0})$ is attached to Levi subgroup which is a product of $GL$-factors (and maybe a symplectic group of smaller rank) such that there is a factor $GL_s$ such that $s\ge 2,$ then there is a set of roots $\theta _0',$ conjugated to $\theta _0,$ such that the factor $GL_s$ is the first factor in the Levi subgroup attached to $\theta _0'.$ This means that $e_{n-n_0+1}-e_{n-n_0+2}$ belongs to $\theta _0'.$ But if we assume that $\sigma (\phi _0)$ (so also $\sigma (\phi ')$) has a non-zero constant term along $\theta _0,$ it has a non-zero constant term along $\theta _0',$ so we can immediately make this assumption about the structure of $\theta _0.$ Now, we apply Lemma 5.2 to conclude that $w$ acts as identity on $\Delta (Sp_{2n_0}),$ so we cannot have $w(\chi _j\nu ^{s_j})=\mu _t\nu ^{\frac{i-1}{2}}.$ On the other hand, if $\theta _0$ is attached to Levi subgroup isomorphic to a product of $GL_1$’s and symplectic group of smaller rank, we have (according to Theorem 5.1)

$$\begin{aligned} \sigma (\phi _0)\hookrightarrow \mathrm{Ind}_{P_{\theta _0}(\mathbb {A})}^{Sp_{2n_0}(\mathbb {A})}(\pi _0'')=\mathrm{Ind}_{P_{\theta _0}(\mathbb {A})}^{Sp_{2n_0}(\mathbb {A})}(\chi _1\nu ^{s_1}\otimes \cdots \otimes \chi _{k_1}\nu ^{s_{k}}\otimes \sigma _0), \end{aligned}$$

(21)

where $\sigma _0$ is a cuspidal automorphic representation of some $Sp_{2m_0}$ and $\chi _1,\ldots , \chi _{k_1}$ are Grossencharacters and $s_i\in \mathbb {R}.$ Since $\sigma (\phi _0)$ is attached to an unipotent Arthur parameter $\phi _0,$ we conclude that

$$\begin{aligned} |s_r|\le \mathrm{max}\left\{ \frac{a_{h,1}-1}{2};1\le h\le l\right\} <\frac{i-1}{2}, \quad \forall r=1,\ldots ,k, \end{aligned}$$

by Conditions ($\Delta$) 1. so we also cannot have $s_j=\frac{i-1}{2}.$ This means, that, if $w=p\varepsilon$, then either $p(i)=i$ or $p(1)=i.$ In the first case, this means $j=i$ and $w=1$, in the second case $j=0$ and $w=w_0,$ as claimed. $\square$

Using the previous proposition we prove that the (meromorphic continuation of) Eisenstein series $E(s,f_s),$ for $f_s$ in a certain (irreducible) subspace of $\pi _s=\mathrm{Ind}_{P_{\Delta {\setminus }\{\alpha _i\}}(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi '))$ (for $s=0$) is non-zero, and as such, gives a realization of this representation in the space of automorphic forms $A(Sp_{2n'+2a_{t,j}}(k){\setminus } Sp_{2n'+2a_{t,j}}(\mathbb {A})).$ Namely, using Theorem 5.1 we prove that the constant term of the image (of this irreducible subspace of $\pi _s$ for $s=0$) by Eisenstein series in non-zero.

Let $f_s=\otimes _v f_{s,v}\in \pi _s=\mathrm{Ind}_{P_{\Delta {\setminus }\{\alpha _i\}}(\mathbb {A})}^{Sp_{2n'+2a_{t,j}}(\mathbb {A})}(|\mathrm{det}|^s\mu _{t,GL(a_{t,j},\mathbb {A})}\otimes \sigma (\phi '))\cong \otimes _v \pi _{s,v}$ where for almost all $v,$ $\pi _{s,v}$ is an unramified representation, and $f_{s,v}$ is $K_v$-invariant vector in $\pi _{s,v},$ normalized in such a way that $f_{s,v}(e_v)=1.$ By Proposition 5.4, $f_s$ and $M(w_0,\pi ,s)f_s$ belong to the same space for $s=0$ (and by the same Proposition the other $M(w,\pi ,s)f_s$ do not belong to the same space) so to prove that $E_{P_{\theta _0}}(s,f_s)(g)$ is non-zero, it is enough to prove $f_s+M(w_0,\pi ,s)f_s$ is not zero. Let $S$ be a finite set of places such that for $v\notin S,$ $\pi _{s,v}$ is unramified. Then, analogously as in the third section, we normalize local intertwining operators. If $v\notin S$ then $\pi _{s,v}$ is a principal series representation, so we can use the normalization introduced in the third section; in this case

$$\begin{aligned} \pi _{s,v}\hookrightarrow \mathrm{Ind}(\mu _{t,v} \nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _{t,v} \nu ^{s+\frac{i-1}{2}}\otimes \lambda _{1,v}\nu ^{s_1}\otimes \cdots \otimes \lambda _{k,v}\nu ^{s_k}\rtimes \sigma (\phi _0)_v) \end{aligned}$$

(22)

(we use the notation from the proof of Proposition 5.4). Now, $\sigma (\phi _0)_v$ is (negative) unramified representation of $Sp_{2n_0}(k_v)$ corresponding to the local unipotent Arthur parameter $\phi _{0,v}.$ Although there does not exist (global) character $\lambda _{\phi ',s,a_{t,j},w_0}$ such that $\sigma (\phi _0)_v$, or more precisely, $\pi _{s,v}$ is embedded in $\mathrm{Ind}(\lambda _{\phi ',s,a_{t,j},w_0,v}),$ we can still easily express the normalizing factor analogous to the normalizing factor $r(\lambda _{\phi ',s,a_{t,j},w_0,v}),$ as we did in the third section (where our representation $\pi _s$ was a subrepresentation globally induced from character). Later, we check that these normalizations give normalized operators which still have good properties, even for $v\in S$ (where locally, we might not have subquotients of principal series representations).

Indeed, assume that $\pi _{s,v}$ is unramified representation, so that $\sigma (\phi _0)_v$ is also unramified and a subquotient of a principal series representation; let us say

$$\begin{aligned} \sigma (\phi _0)_v\hookrightarrow \chi _1\nu ^{s_1'}\times \chi _2\nu ^{s_2'}\times \cdots \times \chi _r\nu ^{s_r'}\rtimes 1. \end{aligned}$$

(23)

The (local) normalizing factor we use in this situation is (according to the third section)

$$\begin{aligned} r_v(s,w_0)=\prod _{\begin{array}{c} \alpha \in \Sigma {^+}\\ w_0(\alpha )<0 \end{array}}\frac{L(1,\lambda _v(s)\circ \check{\alpha })\varepsilon (0,\lambda _v(s)\circ \check{\alpha },\psi _v)}{L(0,\lambda _v(s)\circ \check{\alpha })}. \end{aligned}$$

Here $\lambda _v(s)$ is obvious local character obtained from (22) and (23); we can take $\varepsilon (0,\lambda _v(s)\circ \check{\alpha })=1,$ because we are in the unramified situation. The set of $\alpha \in \Sigma {^+},\;w_0(\alpha )<0$ is divided into four subsets (as in the proof of Lemma 4.2). The contribution (this time locally) from the first and the second set of roots is the same as in (15) and (16). As for the third and the fourth group of roots, we can divide them into two subsets-the first related to the (localization of) global characters $\lambda _1,\ldots ,\lambda _k$ (in (22)) and the other subset-local characters $\chi _1,\ldots , \chi _r$ appearing in (23). While the first subset is obviously related to the global characters, we can also relate (global) Hecke $L$-functions with the second subset of roots; namely if exponents $s_1',\ldots ,s_r'$ appear in (23), they appear in the local $L$ function in the third group of roots, and $-s_1',\ldots ,-s_r'$ appear in the fourth group of roots. This means, that, if we examine the third and the fourth group of roots together (and the corresponding local $L$-functions in the normalizations) it does not matter what exact kind of embedding (23) we have, i.e., only what matters is a cuspidal support of $\sigma (\phi _0)_v.$ We can relate the cuspidal support of that (negative) unramified representation with the Jordan block. For simplicity of notation, assume that $v$ is unramified place such that $\mu _{1,v}=\mu _{2,v}=\cdots =\mu _{r,v}=\chi _0,$ and $\mu _{r+1,v}=\cdots =\mu _{l,v}=1$ (here we use the notation for the basis of our inductive procedure given in the third section). Then

$$\begin{aligned}&\sigma (\phi _0)_v \hookrightarrow \zeta \left( -\frac{a_{1,1}-1}{2},\frac{a_{2,1}-1}{2};\chi _0\right) \times \cdots \times \zeta \left( -\frac{a_{r-1,1}-1}{2},\frac{a_{r,1}-1}{2};\chi _0\right) \nonumber \\&\quad \times \zeta \left( -\frac{a_{r+1,1}-1}{2},\frac{a_{r+2,1}-1}{2};1\right) \times \cdots \zeta \left( -\frac{a_{l,1}-1}{2},-1;1\right) \rtimes 1. \end{aligned}$$

(24)

Then, part of the appropriate normalizing factor (attached to the third and fourth set of the roots) is

$$\begin{aligned}&\prod _{1\le k\le i}\left( \prod _{f=1}^l\prod _{p=-\frac{a_{f,1}-1}{2}}^{\frac{a_{f,1}-1}{2}}\frac{L(-\frac{i-1}{2}+s+k-p,\mu _{t,v}\mu _{f,v})\varepsilon (-\frac{i-1}{2}+s+k-p-1,\mu _{t,v}\mu _{f,v})}{L(-\frac{i-1}{2}+s+k-p-1,\mu _{t,v}\mu _{f,v})}\right) \nonumber \\&\quad \frac{L(-\frac{i-1}{2}+s+k-1,\mu _{t,v})}{L(-\frac{i-1}{2}+s+k,\mu _{t,v})\varepsilon (-\frac{i-1}{2}+s+k-1,\mu _{t,v})}. \end{aligned}$$

The second line above comes from the fact that in the first line we calculated $\nu ^01_{GL_1}$ as a part of cuspidal support in (24), and it isn’t, so we had to divide it by the factor in the second line. Note that this second line exactly cancels with the contribution from the first set of roots in the proof of Lemma 4.2. So, we can conclude

$$\begin{aligned}&r_v(s,w_0)=\prod _{1\le k< l\le i}\frac{L(2s-i+k+l,1_v)\varepsilon (2s-i+k+l-1,1_v,\psi _v)}{L(2s-i+k+l-1,1_v)} \nonumber \\&\quad \prod _{1\le k\le i}\left( \prod _{(\mu ,a)\in \mathrm{Jord}(\phi ')}\prod _{p=-\frac{a-1}{2}}^{\frac{a-1}{2}}\frac{L(-\frac{i-1}{2}+s+k-p,\mu _{t,v}\mu _{v})\varepsilon (-\frac{i-1}{2}+s+k-p-1,\mu _{t,v}\mu _{v})}{L(-\frac{i-1}{2}+s+k-p-1,\mu _{t,v}\mu _{v})}\right) .\nonumber \\ \end{aligned}$$

(25)

This leads us to define

$$\begin{aligned}&r(s,w_0)=\prod _{v}r_v(s,w_0)=\prod _{1\le k< l\le i}\frac{L(2s-i+k+l,1)\varepsilon (2s-i+k+l-1,1)}{L(2s-i+k+l-1,1)}\nonumber \\&\quad \prod _{1\le k\le i}\left( \prod _{(\mu ,a)\in \mathrm{Jord}(\phi ')}\prod _{p=-\frac{a-1}{2}}^{\frac{a-1}{2}}\frac{L(-\frac{i-1}{2}+s+k-p,\mu _{t}\mu )\varepsilon (-\frac{i-1}{2}+s+k-p-1,\mu _{t}\mu )}{L(-\frac{i-1}{2}+s+k-p-1,\mu _{t}\mu )}\right) . \end{aligned}$$

although this normalization is justified (for now) only for $v\notin S.$

Continuing after Theorem 5.3 and Proposition 5.4, and the discussion after it, we conclude the following (for $f_s=\otimes f_{s,v}\in \pi _s$):

$$\begin{aligned} M(w_0,\pi ,s)f_s&=r(s,w_0)^{-1}(\otimes _{v\notin S}r_v(s,w_0)A(w_0,\pi _v,s)f_{v,s}\otimes _{v\in S} r_v(s,w_0)A(w_0,\pi _v,s)f_{v,s})\\&=r(s,w_0)^{-1}(\otimes _{v\notin S}N(w_0,\pi _v,s)f_{v,s}\otimes _{v\in S}N(w_0,\pi _v,s)f_{v,s}), \end{aligned}$$

where we have defined normalized operators as in the Sect. 4.1. (for $v\notin S$). So, for $v\notin S$ we can define subspaces $Y_v(\sigma (\phi '))^{\pm }$ as in Sect. 4.1. We now prove that for $v\in S$, the normalized operators also have the required properties.

Proposition 5.5

With the notation as above, $\lim _{s\rightarrow 0}r(s,w_0)^{-1}=1.$ For $v \in S$ the intertwining operator $N(w_0,\pi _v,s)$ is holomorphic for $s=0,$ and the space $\mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma (\phi ')_v)$ decomposes into two subspaces $Y_v(\sigma (\phi '))^+$ and $Y_v(\sigma (\phi '))^-,$ where this operators acts as identity and minus identity, respectively.

Proof

Using the global functional equation for Hecke Grossencharacters and change of variables, similarly as in the proof of Lemma 4.2, we get that

$$\begin{aligned} \lim _{s\rightarrow 0}r(s,w_0)^{-1}=1. \end{aligned}$$

Assume that $v\in S$ is a non-archimedean place. We briefly recall a certain involution on the set of (local) Arthur parameters for the classical groups (for the general definition of the Arthur parameters we refer to [1]). If $\psi$ is some (local) Arthur parameter $\psi :W_{k_v}\times SL(2,\mathbb {C})\times SL(2,\mathbb {C})\rightarrow SO(2n+1,\mathbb {C}),$ $\hat{\psi }$ is Arthur parameter obtained by interchanging the action of two $SL(2,\mathbb {C})$’ s. The Arthur parameter is generic if it is trivial on the second copy of $SL(2,\mathbb {C})$ and it then corresponds to a tempered packet; in this paper we work with the unipotent parameter; i.e., trivial on the first $SL(2,\mathbb {C}).$ By the results of Mœglin [13], which are partially global, depending on Arthur’s Theorem 2.2.1 of [1], (cf. [1, section 7.1]), we know that when we apply involution on the generic Arthur parameter (to obtain a unipotent parameter), the associated representations in the unipotent Arthur packets are (Aubert–Schneider–Schuler) duals of the representations in the original generic Arthur packets. For an irreducible representation $\sigma ,$ we denote by $\hat{\sigma }$ its Aubert dual representation ($\hat{\sigma }$ is a genuine representation, so it is plus or minus Aubert dual representation as defined in [3]). Note that, in that case, in our notation, ${\sigma (\hat{\phi _v'})}$ describes a tempered packet, and the Plancherel measure attached to the induced representation $\mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(|\mathrm{det}|_v^s\mu _{t,v}\mathrm{St}_{GL_{a_{t,j},k_v}}\otimes \hat{\sigma (\phi ')_v})$ can be calculated in terms of this (Arthur) parameter (i.e., Jordan block, cf. [14, Section 13]). Here $\mathrm{St}_{GL_{a_{t,j},k_v}}$ denotes the Steinberg representation of $GL_{a_{t,j}}(k_v),$ which is the Aubert dual of the trivial representation of $GL_{a_{t,j}}(k_v).$ The results of Ban [4, the proof of Lemma 7.1], cf. [1, section 7.1] (obtained without any restrictions) show that the Plancherel measure $\mu (s,\mu _{t,v}\mathrm{St}_{GL_{a_{t,j},k_v}}\otimes \hat{\sigma (\phi ')_v})$ attached to the representation $\mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(|\mathrm{det}|_v^s\mu _{t,v}\mathrm{St}_{GL_{a_{t,j},k_v}}\otimes \hat{\sigma (\phi ')_v})$ is the same as the Plancherel measure attached to the representation

$$\begin{aligned} \mathrm{Ind}_{P(k_v)}^{Sp_{2n'+2a_{t,j}}(k_v)}(|\mathrm{det}|_v^s\mu _{t,GL(a_{t,j},k_v)}\otimes \sigma (\phi ')_v) \end{aligned}$$

(here we have a particular representation $\sigma (\phi ')_v$ belonging to the local Arthur packet, but the measure should be the same for any member of the packet). Because we are interested in the intertwining operator attached to the longest element $w_0$ of the Weyl group, we prove that (5-18) and (5-19) of [18] hold, and this is enough to prove the claim of the Proposition. By the above discussion, to prove these relations it is enough to show that

$$\begin{aligned} r_v(s,w_0)r_v(-s,w_0)= \mu (s,\mu _{t,v}\mathrm{St}_{GL_{a_{t,j},k_v}}\otimes \hat{\sigma (\phi ')_v}). \end{aligned}$$

We can use [14, Section 13] to easily obtain the above result, but only up to a non-zero constant (since [14] gives the expression for the Plancherel measure up to a non-zero constant). After adjusting our normalization by this constant, we get (5–19) of [18] and we can introduce spaces $Y_v(\sigma (\phi '))^{\pm }.$ This constant is not a serious obstacle, since it is a positive real number (we know that $\mu (0,\mu _{t,v}\mathrm{St}_{GL_{a_{t,j},k_v}}\otimes \hat{\sigma (\phi ')_v})\ge 0$ and we assume $\overline{\psi _v}=\psi _v$) so introducing spaces $Y_v(\sigma (\phi '))^{\pm }$ still makes sense (even without second normalization with this constant).

Assume $v\in S$ is an archimedean place. We note that the normalization of the intertwining operators attached to the local Arthur parameter $\psi _v$ (we use this notation for $\phi _v$ to avoid confusion) explained in the section 2.3 of [1] is uniform (used for all the members of (non-generic) packet $\psi _v$) and obtained in the following way: we introduce $\phi _{\psi _v}:W_{k_v}\rightarrow SO(2n+1,\mathbb {C})$ given by . We easily see that, for a representation $\psi _v=\mu _v\otimes V_a$ of $W_{k_v}\times SL(2,\mathbb {C}),$ its pull back $\psi _{\phi _v}$ to $W_{k_v}$ equals $\oplus _{k=0}^{a-1}\mu _v(\cdot )|\cdot |^{-\frac{a-1}{2}+i}.$ When we, in our situation, use then formulas (2.3.3), (2.3.8) and (2.3.26) and (2.3.27) of [1] (to be in the same situation as in our case), calculate these normalizing factors, we obtain the same formulas as we have for $r_v(w_0,s).$ Then Proposition 2.3.1 of [1] guarantees that our normalization satisfies the claim of this Proposition. $\square$

Proposition 5.6

Let $S$ be a finite set of places, containing all the archimedean places and if $v\notin S,\,\pi _{0,v}$ is unramified (cf. (19)). Assume that $|S|$ is even. Let $\pi$ be an irreducible subrepresentation of the representation $\otimes _{v\notin S}Y_v(\sigma (\phi '))^{+}\otimes _{v\in S} Y_v(\sigma (\phi '))^{-}.$ Then, the Eisenstein series from Theorem 5.3 acting on the representation $\pi _s$ is non-zero on $\pi$ for $s=0.$

Proof

By the the discussion before this proposition, we saw that, for the choice $f_s=\otimes f_{s,v}$ from $\pi$, the following holds: $\lim _{s\rightarrow 0}M(w_0,\pi ,s)f_s=Cf_0,$ where $C>0$ so that $\lim _{s\rightarrow 0}E_{P_{\theta _0}}(s,f_s)\ne 0,$ and, consequently, $\lim _{s\rightarrow 0}E(s,f_s)\ne 0.$ $\square$

5.2 Step 2

We now continue our construction, and we have the analogon of Proposition 4.3. The image of $\pi$ under the action of the Eisenstein series is denoted by $E(\pi ).$ We denote the ṡpace of constant terms of $E(\pi )$ along $P_{\theta _0}$ by $E_{\theta _0}(\pi ),$ (in the previous proposition we proved that it is non-zero). We study the induced representation $\mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\mu _t|det|^s\otimes E_{\theta _0}(\pi )),$ and calculate Eisenstein series associated to this representation: $f_s\mapsto E(s,f_s)(g)=\sum _{\gamma \in P(k){\setminus } Sp_{2n}(k)}f_s(\gamma g),$ where $P$ is a maximal parabolic subgroup (analogous to the one in Proposition 4.3. We prove the following proposition:

Proposition 5.7

The map

$$\begin{aligned} \pi _s=\mathrm{Ind}_{P(\mathbb {A})}^{Sp_{2n}(\mathbb {A})}(\mu _t|det|^{\frac{a_{t,j}+a_{t,j+1}}{4}}\otimes E_0(\pi ))\rightarrow A(Sp_{2n}(k){\setminus } Sp_{2n}(\mathbb {A})) \end{aligned}$$

given by

$$\begin{aligned} f_{{\frac{a_{t,j}+a_{t,j+1}}{4}}}\rightarrow \left( s-{\frac{a_{t,j}+a_{t,j+1}}{4}}\right) ^2E(s,f_s)(\cdot )|_{s={\frac{a_{t,j}+a_{t,j+1}}{4}}} \end{aligned}$$

is well-defined and non-trivial, and its image $E(\pi ,\mu _t,a_{t,j+1})$ is contained in the space of square-integrable automorphic forms. Every irreducible subrepresentation of $E(\pi ,\mu _t,a_{t,j+1})$ (which is semi-simple representation) is thus automorphic square-integrable, and has a global Arthur parameter equal to $\phi .$

Proof

Proof of this proposition will differ somewhat from the proof of Proposition 4.3. Again, to prove that the Eisenstein series has a pole of order two, we calculate the order of a pole for the constant term $E_{P_{\theta _0}}(s,f_s)(g)$ (we recall our convention about the notation of roots $\theta _0$). Again, as in the proof of Theorem 5.3, for $f_s=\otimes f_{s,v}$ we have

$$\begin{aligned} E_{P_{\theta _0}}(s,f_s)(g)=\sum _{\begin{array}{c} w \in W,\,w(\Delta {\setminus } \{\alpha \})>0\\ w(\theta _0)=\theta _0 \end{array}}(M(w,\pi ,s)f_s)(g). \end{aligned}$$

(26)

Let $S$ be a finite set of places, such that for $v\notin S\,\pi _{s,v}$ is unramified and $f_{s,v}$ is a spherical vector, normalized as usual. For $v\notin S$, let $\chi _{s,v}$ be a character of maximal torus in the symplectic group in question, such that $\pi _{s,v}$ is embedded in the principal series representation induced by the character $\chi _{s,v}.$ Assume that for those $v,$ $\psi _v$ is unramified. We then have

$$\begin{aligned} M(w,\pi ,s)f_s=\otimes _{v\notin S}\prod _{\begin{array}{c} \beta >0\\ w(\beta )<0 \end{array}}\frac{L(0,\chi _{s,v}\circ \check{\beta })}{L(1,\chi _{s,v}\circ \check{\beta })}f_{w(s),v}\otimes _{v\in S} A(w,s,\pi _v)f_{s,v}. \end{aligned}$$

(27)

Note that now $i=\frac{a_{t,j+1}-a_{t,j}}{2},$ so for $v\notin S,$ $s$ near $\frac{a_{t,j+1}+a_{t,j}}{4}$

$$\begin{aligned}&\chi _{s,v}=\mu _{t,v}\nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _{t,v}\nu ^{s+\frac{i-1}{2}}\otimes \mu _{t,v}\nu ^{-\frac{a_{t,j}-1}{2}}\otimes \cdots \otimes \mu _{t,v}\nu ^{\frac{a_{t,j}-1}{2}}\otimes \lambda _{1,v}\nu ^{s_1}\nonumber \\&\quad \otimes \cdots \otimes \lambda _{k,v}\nu ^{s_k}\rtimes \sigma (\phi _0)_v), \end{aligned}$$

cf. (22).

The difference in the proof of this proposition and Proposition 4.3 is in the fact that we calculate the order of pole of $M(w,\pi ,s)f_s$ by calculating the orders of the poles of partial Hecke $L$-functions obtained from the previous expression. Although the representation $\pi _s$ is not globally induced from Hecke characters, in the same way as in the discussion after Proposition 5.4, we can attach to it (partial $L$-functions of) a certain global character. As we saw in the proof of Lemma 4.4, that for given $w\in W$ as above, the set of all $\beta >0$ such that $w(\beta )<0$ can be divided in the six sets. We examine the behavior of the partial $L$-functions which we form when we explicitly express $\prod _{v \notin S}\frac{L(0,\chi _{s,v}\circ \check{\beta })}{L(1,\chi _{s,v}\circ \check{\beta })}$ for every set of roots.

The contribution from the first set of roots (where $w=p\epsilon ,\;w \in W_j$) is

$$\begin{aligned} \prod _{k=j+1}^i\frac{L_S\left( -\frac{i-1}{2}+s+k-1,\mu _t\right) }{L_S\left( -\frac{i-1}{2}+s+k-1,\mu _t\right) }. \end{aligned}$$

Note that for $s\approx \frac{a_{t,j+1}+a_{t,j}}{4}$ every factor in the numerator (and denominator) is strictly positive. Note that in the integer points (the full) Hecke $L$-functions are non-zero. Also note that in the strictly positive points, local Hecke $L$-functions (attached to quadratic characters) are non-zero and do not have a pole. We conclude that this contribution has no zeros or poles, except when $\mu _t=1,\;a_{t,j}=1,\;j=0,$ when $L_S(1,\mu _t)$ has a pole of the first order (appearing in the numerator), so the whole contribution has a pole of the first order.

The contribution from the second set of roots is given by

$$\begin{aligned} \prod _{j+1\le k<l\le i}\frac{L_S(2s-i+k+l-1;1)}{L_S(2s-i+k+l;1)}. \end{aligned}$$

As $2s-i\approx a_{t,j}$ we see that $k+l-1\ge 3,$ so this expression has no zeros or poles; and if $j=i$ or $j+1=i$ there is no second set of roots at all.

Analogously, we see that the fourth set of roots contributes with expression which has no zeros or poles, and there is no contribution if $j=0$ or $j=i.$

The third set of roots does not exist if $j=i.$ Otherwise, we can divide the expression

$$\begin{aligned} \prod _{\begin{array}{c} j+1\le k\le i\\ i+1\le l\le n \end{array}}\otimes _{v\notin S}\frac{L(0,\chi _{s,v}\circ (\check{e_k}-\check{e_l}))}{L(1,\chi _{s,v}\circ (\check{e_k}-\check{e_l}))} \end{aligned}$$

into a product of three factors, according whether

• $j+1\le l\le \frac{a_{t,j+1}+a_{t,j}}{2}$ (case (a)),

• $\frac{a_{t,j+1}+a_{t,j}}{2}+1\le l\le n-n_0$ (case (b)) and

• $n-n_0+1\le l\le n$ (case (c)).

Contribution from case (a) equals

$$\begin{aligned} \prod _{\begin{array}{c} j+1\le k\le i\\ i+1\le l\le \frac{a_{t,j+1}+a_{t,j}}{2} \end{array}}\frac{L\left( s-\frac{i-1}{2}+\frac{a_{t,j}-1}{2}+k-l+i;1\right) }{L\left( s-\frac{i-1}{2}+\frac{a_{t,j}-1}{2}+k-l+i+1;1\right) }. \end{aligned}$$

If $j=0$ then for $k=1$ and $l=\frac{a_{t,j+1}+a_{t,j}}{2}$ there is a factor $L_S(1;1)$ in the numerator; if $j\ge 1$ there are no poles or zeros in the numerator or denominator. So, if $j=0$ there is a pole of the first order. As for the case (b), we easily again get global (partial) Hecke functions, like in the case (a), and by our assumption ($\Delta$) on the way of adding new elements in Jordan block ($a_{t,j}$ is greater or equal to every previous member of Jordan block (associated to any appearing character)) we can conclude that there is no zeros or poles in that contribution. The case (c) is not directly related to the global characters, but we can go around it in the following way: according to the proof of Proposition 5.4, either $w(\Delta (Sp_{2n_0}))=\mathrm{id}$ or for $\sigma (\phi _0)$ relation (21) holds. In the first case we can automatically conclude that $p(l)>p(k)$ for case (c) ($p(l)=l$), but then we can combine this case with the contribution of the sixth group of roots (case (c); we divide this group of roots in cases in the same way as for the third group) and analogously as in the discussion after Proposition 5.4 we get the contribution

$$\begin{aligned} \prod _{j+1\le k\le i}\left( \prod _{r=1}^l\prod _{p=-\frac{a_{r,1}-1}{2}}^{\frac{a_{r,1}-1}{2}}\frac{L_S\left( s\!-\!\frac{i-1}{2}\!+\!k\!-\!1\!-\!p,\mu _t\mu _r\right) }{L_S\left( s-\frac{i-1}{2}\!+\!k\!-\!p,\mu _t\mu _r\right) } \right) \frac{L_S\left( s\!-\!\frac{i-1}{2}\!+\!k,\mu _t\right) }{L_S\left( s-\frac{i-1}{2}\!+\!k\!-\!1,\mu _t\right) }. \end{aligned}$$

We easily see that this expression has no zeros or poles unless $\mu _t=1,\;a_{t,j}=1,\;j=0$ when it has a zero of the first order. If (21) holds for $\sigma (\phi _0),$ we proceed as follows: we divide the contribution 3 (c) into subsets

$$\begin{aligned} \prod _{\begin{array}{c} j+1\le k\le i\\ n-n_0+1\le l\le n-m_0\\ p(l)>p(k) \end{array}}\frac{L_S\left( s-\frac{i-1}{2}+k-1-s_{l-(n-n_0)};\mu _t\chi _{l-(n-n_0)}\right) }{L_S\left( s-\frac{i-1}{2}-s_{l-(n-n_0)};\mu _t\chi _{l-(n-n_0)}\right) },\\ \prod _{\begin{array}{c} j+1\le k\le i\\ n-n_0+1\le l\le n-m_0\\ p(l)<p(k) \end{array}}\frac{L_S\left( s-\frac{i-1}{2}+k-1-s_{l-(n-n_0)};\mu _t\chi _{l-(n-n_0)}\right) }{L_S\left( s-\frac{i-1}{2}-s_{l-(n-n_0)};\mu _t\chi _{l-(n-n_0)}\right) } \end{aligned}$$

and, the last contribution, coming from embedding $\sigma _{0,v}$ in principal series (cf. Theorem 2.2 and Corollary 2.3), which we combine with analogous situation for the part of sixth (c) case (since here $p(l)=l>p(k)$):

$$\begin{aligned}&\prod _{j+1\le k\le i}\left( \prod _{r=1}^l\prod _{p=-\frac{a_{r,1}-1}{2}}^{\frac{a_{r,1}-1}{2}}\frac{L_S\left( s-\frac{i-1}{2}+k-1-p,\mu _t\mu _r\right) }{L_S\left( s-\frac{i-1}{2}+k-p,\mu _t\mu _r\right) } \right) \frac{L_S\left( s-\frac{i-1}{2}+k,\mu _t\right) }{L_S\left( s-\frac{i-1}{2}+k-1,\mu _t\right) }\nonumber \\&\quad \prod _{r=1}^{k_1}\frac{L_S\left( s-\frac{i-1}{2}+k-s_r,\mu _t\chi _r\right) L_S\left( s-\frac{i-1}{2}+k+s_r,\mu _t\chi _r\right) }{L_S\left( s-\frac{i-1}{2}+k-1-s_r,\mu _t\chi _r\right) L_S\left( s-\frac{i-1}{2}+k-1+s_r,\mu _t\chi _r\right) }. \end{aligned}$$

Again, the result is the same as for the situation $w(\Delta (Sp_{2n_0}))=\mathrm{id}.$

As for the fifth set of roots (the contribution is not there if $j=0$): again, we divide the contribution in three cases, as above. The contribution of the case (a) is

$$\begin{aligned} \prod _{\begin{array}{c} 1\le k\le j\\ i+1 \le l\le \frac{a_{t,j+1}+a_{t,j}}{2}\\ p(l)<p(k) \end{array}}\frac{L_S\left( s-\frac{i-1}{2}+k-1+\frac{a_{t,j}-1}{2}-l+(i-1);1\right) }{L_S\left( s-\frac{i-1}{2}+k+\frac{a_{t,j}-1}{2}-l+(i-1);1\right) }. \end{aligned}$$

We see that this expression has a pole of the first order if $k=1,\; l=\frac{a_{t,j+1}+a_{t,j}}{2}$ (and ($p(\frac{a_{t,j+1}+a_{t,j}}{2})<p(1)$). As for the contribution from the case $b$ ($l$ varies among the exponents attached to the adding new elements in Jordan blocks), it easily follows that, with the assumption ($\Delta$), it does not have zeros or poles. The discussion in the case (c) again resembles the discussion about the third set of roots: if $w$ acts as an identity on $\Delta (Sp_{2n_0}),$ then there is no contribution from this set of roots, since we cannot have $l=p(l)<p(k)$ (since ($n-n_0+1\le l\le n$)). If, on the other hand, we have (21), then if $p(l)<p(k)$ for roots in the case (c), then they are attached to characters $\chi _\nu ^{s_1},\ldots ,\chi _k\nu ^{s_k}$ in the notation of (21), so our assumption ($\Delta$) then guarantees that the contribution from this roots does not have zeros or poles (analogously to the case (b)).

We are left to analyze the sixth set of roots (case (a) and (b) ; case (c) is resolved). The contribution from the case (a) is

$$\begin{aligned} \prod _{\begin{array}{c} j+1\le k\le i\\ i+1\le l \le \frac{a_{t,j+1}+a_{t,j}}{2}\\ p(l)>p(k) \end{array}}\frac{L_S\left( s-\frac{i-1}{2}+k-1-\frac{a_{t,j}-1}{2}+l-(i+1);1\right) }{L_S\left( s-\frac{i-1}{2}+k-\frac{a_{t,j}-1}{2}+l-(i+1);1\right) }. \end{aligned}$$

We note that for $j=0,\;k=1$ and $l=i+1$ we obtain factor $L_S(1;1)$ in the numerator if $p(i+1)>p(1)$ (and this is the only possible pole). There is no pole if $j\ge 1.$ Again assuming ($\Delta ),$ the contribution from case (b) is without zeros and poles.

We conclude: if $j\ge 1$ the expression $\otimes _{v \notin S}A(w,s,\pi _v)f_{s,v}$ has a pole of the first order (if additionally $p(1)>p(\frac{a_{t,j+1}+a_{t,j}}{2})$). If $j=0,$ this expression has a pole of the first order or the pole of second order if $p(i+1)>p(1).$ But the last condition on $w\in W$ says that there is only one such $w,$ namely $w=w_0.$ To conclude: $\otimes _{v \notin S}A(w,s,\pi _v)f_{s,v}$ for $w\ne w_0$ has a pole of at most the first order for $s=\frac{a_{t,j+1}+a_{t,j}}{4}$ and for $w=w_0$ it has a pole of the second order.

Now we analyze $\otimes _{v\in S}A(w,s,\pi _v)f_{s,v}.$ We prove that all these operators are holomorphic for $s=\frac{a_{t,j+1}+a_{t,j}}{4}.$ This, in turn, proves that the mapping $f_{{\frac{a_{t,j}+a_{t,j+1}}{4}}}\mapsto (s-{\frac{a_{t,j}+a_{t,j+1}}{4}})^2E(f_s,\cdot )|_{s={\frac{a_{t,j}+a_{t,j+1}}{4}}}$ is well-defined and non-trivial (cf. (26), (27)). Recall that

$$\begin{aligned}&\pi _{s,v}\hookrightarrow \mathrm{Ind}_{P_{\theta _0}(k_v)}^{Sp_{2n}(k_v)}(\mu _{t,v}\nu ^{s-\frac{i-1}{2}}\otimes \cdots \otimes \mu _{t,v}\nu ^{s+\frac{i-1}{2}}\otimes \mu _{t,v}\nu ^{-\frac{a_{t,j}-1}{2}}\otimes \cdots \\&\quad \otimes \mu _{t,v}\nu ^{\frac{a_{t,j}-1}{2}}\otimes \lambda _{1,v}\nu ^{s_1}\otimes \cdots \otimes \lambda _{k,v}\nu ^{s_k}\rtimes \sigma (\phi _0)_v)), \end{aligned}$$

where $\sigma (\phi _0)_v$ is not (necessarily) a principal series subquotient, and we decompose each $A(w,s,\pi _v)$ into a product of generalized rank-one intertwining operators corresponding to the decomposition of $w$ (cf. Lemma 2.1.2 and Theorem 2.1.1 of [20]). So it is either study of $GL_2$-case (with Levi isomorphic to $GL_1\times GL_1$) or study of the intertwining operator acting on a maximal Levi subgroup $GL_1\times Sp_{2n_0}$ (because $w(\theta _0)=\theta _0$). The exact description of the action of the elements of the Weyl group is given, for example, in [17, (3.15) and (3.16)], where the action is first described for certain elements $w_j \in W_j,$ and then the action of all other elements in $W_j$ is described using “shuffles”, as is explained in loc.cit). According to this description, we are only interested (in $GL_1\times GL_1$ case) in intertwining operator $\chi _1\times \chi _2\rightarrow \chi _2\times \chi _1,$ where $\chi _1$ is one of the characters $\mu _{t,v}\nu ^{\pm (s-\frac{i-1}{2})}, \ldots ,\mu _{t,v}\nu ^{\pm (s+\frac{i-1}{2})},$ and $\chi _2$ one of the characters $\mu _{t,v}\nu ^{-\frac{a_{t,j}-1}{2}},\ldots , \mu _{t,v}\nu ^{\frac{a_{t,j}-1}{2}},\lambda _{1,v}\nu ^{s_1},\ldots ,\lambda _{k,v}\nu ^{s_k},$ or vice-versa, or $\chi _1$ is one of the characters $\mu _{t,v}\nu ^{s-\frac{i-1}{2}},\ldots , \mu _{t,v}\nu ^{s-\frac{i-1}{2}+j-1}$ and $\chi _2$ one of the characters $\mu _{t,v}\nu ^{\pm (s-\frac{i-1}{2}+j)},\ldots ,\mu _{t,v}\nu ^{\pm (s+\frac{i-1}{2})}$ (or vice-versa). In both cases we have holomorphy of intertwining operators (for archimedean and non-archimedean) places. Now we are left to prove the holomorphy of intertwining operators

$$\begin{aligned}&\mu _{t,v}\nu ^{s-\frac{i-1}{2}+j}\rtimes \sigma (\phi _0)_v\rightarrow \mu _{t,v}\nu ^{-(s-\frac{i-1}{2}+j)}\rtimes \sigma (\phi _0)_v, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \vdots \nonumber \\&\quad \mu _{t,v}\nu ^{s+\frac{i-1}{2}}\rtimes \sigma (\phi _0)_v \rightarrow \mu _{t,v}\nu ^{-(s+\frac{i-1}{2})}\rtimes \sigma (\phi _0)_v. \end{aligned}$$

(28)

Assume now that $v$ is non-archimedean. We prove that these intertwining operators are holomorphic, by embedding the representation $\sigma (\phi _0)_v$ into induced representation, using the cuspidal support of $\sigma (\phi _0)_v,$ and then these intertwining operators are viewed as the restrictions of the intertwining operators on these induced representations. We actually observe that these (new) intertwining operators are holomorphic. We do that by estimating the cuspidal support of the representation $\sigma (\phi _0)_v.$ The representation $\hat{\sigma (\phi _0)_v}$ is tempered and has the same cuspidal support as $\sigma (\phi _0)_v.$ We recall that, for an irreducible admissible representation $\pi$ of a symplectic group there exists a representation $\pi '$ of an appropriate general linear group and the unique irreducible supercuspidal representation $\pi _{cusp}$ of a symplectic group such that

$$\begin{aligned} \pi \hookrightarrow \pi '\rtimes \pi _{cusp}. \end{aligned}$$

(29)

We call $\pi _{cusp}$ the partial cuspidal support of $\pi .$

If $\pi$ is tempered representation, by the classification of discrete series (and tempered representations) of classical groups by Moœglin and Tadić [14], we can attach to it a finite multiset, called the Jordan block of this tempered representation. The Jordan block consists of pairs consisting of a cuspidal representation of general linear group and a positive rational integer. Together with the partial cuspidal support (and another parameter, called the $\varepsilon$-function), the Jordan block describes how this tempered representation is embedded in a parabolically induced representation, similarly to the way in which Jordan blocks of negative representations describe the embedding of that unramified representation in a principal series representation (cf.(2)). We then define support of the Jordan block of a tempered representation $\pi ,$ denoted by $\mathrm{supp}(\mathrm{Jord}(\pi )),$ in an analogous way to the way in which support of a Jordan block is defined for unramified representations (cf. Definition 2.1), but now the sum in this definition runs through $(\rho ,a)$ belonging to the Jordan block ($\rho$ is an irreducible cupsidal representation, as explained above).

Although we do not know explicitly the partial cuspidal support of $\hat{\sigma (\phi _0)_v}$ just by knowing its Jordan block, we know that

$$\begin{aligned} \mathrm{supp}(\mathrm{Jord}(\hat{\sigma (\phi _0)_v}_{cusp}))\subset \mathrm{supp}(\mathrm{Jord}(\hat{\sigma (\phi _0)_v})). \end{aligned}$$

Form the definition of the Jordan block (more details in [14]) it follows that if an irreducible cuspidal representation of a general linear group, say $\rho ,$ is in the cuspidal support of a representation $\pi '$ in a situation of (29), then $\rho$ is in $\mathrm{supp}(\mathrm{Jord})(\pi ).$ This means that, when we embed $\hat{\sigma (\phi _0)_v}$ in the representation induced from the cuspidal one, all the exponents will be smaller (by an absolute value) of the exponents in $\mu _{t,v}\nu ^{s-\frac{i-1}{2}+j},\ldots ,\mu _{t,v}\nu ^{s+\frac{i-1}{2}}.$ This means that all the intertwining operators in (28) are holomorphic.

For $v\in S$ archimedean we proceed as follows. To some local Arthur parameter $\psi _v$ we can attach parameter $\phi _{\psi _v}$ like in the proof of Proposition 5.5 (so, in our notation, we form the parameter $\phi _{\sigma (\phi )_v}).$ To this new parameter we attach a (non-tempered) packet, like in Proposition 7.4.1 of [1]. In the proof of that proposition, there is a claim which says that every member of Arthur packet $\sigma (\phi _0)_v$ has a (Langlands) parameter smaller (or equal) to the Langlands parameter (or linear form) of $\phi _{\sigma (\phi _0)_v}.$ But exponents in this form vary from $-\frac{a_{k,1}-1}{2},\ldots ,\frac{a_{k,1}-1}{2},\;k=1,\ldots , l,$ and are strictly smaller of exponents in $\mu _{t,v}\nu ^{s-\frac{i-1}{2}+j},\ldots ,\mu _{t,v}\nu ^{s+\frac{i-1}{2}}.$ This means that all the intertwining operators in (28) are holomorphic at archimedean places, too. $\square$

Now the rest of the proof (i.e., the conclusion in which we argue that the representation thus realized in the space of square-integrable automorphic forms $A(Sp_{2n}(k){\setminus } Sp_{2n}(\mathbb {A}))$ really has Arthur parameter equal to $\sigma (\phi )$) is analogous to the argument in the third section.