Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $\mathrm{GL}_{2n}$: Real Case

TL;DR

This paper constructs explicit cohomological vectors and local integrals for $ ext{GL}_{2n}$, advancing the understanding of non-vanishing and cohomological test vectors for standard $L$-functions, with implications for arithmetic applications.

Contribution

It explicitly constructs archimedean local integrals and cohomological vectors that realize the local standard $L$-functions, providing a new approach to non-vanishing hypotheses at infinity.

Findings

Constructed a new twisted linear functional $ ext{Lambda}_{s, ext{chi}}$

Explicitly realized local standard $L$-functions via cohomological vectors

Established non-vanishing results using different methods

Abstract

The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places.

Equations

$${\int }_{[{\mathrm{GL}}_n\times {\mathrm{GL}}_n]}{\phi }_{\pi }(\left(\begin{array}{cc}{g}_1& 0\\ 0& g_2\end{array}\right))|\frac{\det g_1}{\det g_2}{|}^{s-\frac{1}{2}}\chi (\frac{\det g_1}{\det g_2})\eta (\det g_2)d{g}_{1}d{g}_2,$$

$$\mathcal{Z}(V_{{\phi }_{\pi }},\chi ,s):={\int }_{{\mathrm{GL}}_n(\mathbb{A})}{V}_{{\phi }_{\pi }}\left(\begin{array}{cc}g& 0\\ 0& {\mathrm{I}}_n\end{array}\right)\chi (\det g)|\det g{|}^{s-\frac{1}{2}}dg$$

$$s(x,g)=\left(\begin{array}{cc}{\mathrm{I}}_n& x\\ 0& {\mathrm{I}}_n\end{array}\right)\left(\begin{array}{cc}g& 0\\ 0& g\end{array}\right)$$

$$V_{{\phi }_{\pi }}(h):={\int }_{Z_{2n}(\mathbb{A})S(k)\backslash S(\mathbb{A})}{\phi }_{\pi }(s(x,g)h){\theta }_{\eta }(s(x,g))ds.$$

$$\mathcal{Z}(V_{{\phi }_{\pi }},\chi ,s)=\prod _v{\mathcal{Z}}_v(V_{{\phi }_v},{\chi }_v,s)$$

$${\mathcal{Z}}_v(V_{{\phi }_v},{\chi }_v,s):={\int }_{{\mathrm{GL}}_n(k_v)}{V}_{{\phi }_v}\left(\begin{array}{cc}g& 0\\ 0& {\mathrm{I}}_n\end{array}\right){\chi }_v(\det g)|\det g{|}_v^{s-\frac{1}{2}}dg.$$

$${\mathcal{Z}}_v(V_{{\phi }_v},{\chi }_v,s)=L(s,{\pi }_v\otimes {\chi }_v)$$

$$\frac{1}{L(s,{\pi }_v\otimes {\chi }_v)}{\mathcal{Z}}_v(V_{{\phi }_{v,s}},{\chi }_v,s)=1.$$

$$\rho =(\frac{2n-1}{2},\frac{2n-3}{2},\cdots ,\frac{3-2n}{2},\frac{1-2n}{2}).$$

$$H=\Big{\{}\begin{pmatrix}g_{1}&\\ &g_{2}\end{pmatrix}\Big{\lvert}g_{1},g_{2}\in{\mathrm{GL}}_{n}(\mathbb{R})\Big{\}}\simeq{\mathrm{GL}}_{n}(\mathbb{R})\times{\mathrm{GL}}_{n}(\mathbb{R}).$$

$$d=n^2+n-1,$$

$$\nu =({\nu }_1,{\nu }_2,\cdots ,{\nu }_{2n})\in {\mathbb{Z}}^{2n},$$

$${\mathrm{H}}^{\ast }(\mathfrak{g},K^0{Z}^0,\pi \otimes F_{\nu }^{\vee })$$

$${\mathrm{H}}^j(\mathfrak{g},K^0{Z}^0,\pi \otimes F_{\nu }^{\vee })$$

$${\nu }_1+{\nu }_{2n}={\nu }_2+{\nu }_{2n-1}=\cdots ={\nu }_{2n}+{\nu }_1:=m\in \mathbb{Z}.$$

$$n^2\le j\le n^2+n-1.$$

$$\vec{l}:=2\nu +2\rho -(m,m,\cdots ,m),$$

$$l_i={\nu }_i-{\nu }_{2n+1-i}+(2n+1-2i)\text{ for all }1\le i\le 2n.$$

$${\eta }_k=\{\begin{array}{ll}\text{id}& \text{ if k is odd};\\ \text{sgn}& \text{ if k is even}.\end{array}$$

$$\pi \simeq {\mathrm{Ind}}_P^G{D}_{l_1}|{|}^{\frac{m}{2}}\otimes D_{l_2}|{|}^{\frac{m}{2}}\otimes \cdots \otimes D_{l_n}|{|}^{\frac{m}{2}},$$

$${\omega }_{\pi }(a{\mathrm{I}}_{2n})=\{\begin{array}{l}|a{|}^{mn}\text{ if m is even};\\ |a{|}^{mn}(\text{sgn}(a){)}^n\text{ if m is odd}.\end{array}$$

$$\omega (a)=\{\begin{array}{l}|a{|}^m\text{ if m is even};\\ |a{|}^m\text{sgn}(a)\text{ if m is odd}.\end{array}$$

$${\mathrm{H}}^{\ast }(\mathfrak{g},K^0{Z}^0,\pi \otimes F_{\nu }^{\vee })\ne 0.$$

$${\mathrm{Ind}}_P^G{D}_{l_1}|{|}^{\frac{m}{2}}\otimes D_{l_2}|{|}^{\frac{m}{2}}\otimes \cdots \otimes D_{l_n}|{|}^{\frac{m}{2}},$$

$$⟨\lambda ,\pi (\left(\begin{array}{cc}{\mathrm{I}}_{n^{\prime }}& Y\\ & {\mathrm{I}}_{n^{\prime }}\end{array}\right)\left(\begin{array}{cc}g& \\ & g\end{array}\right))v⟩=\tilde{\omega }(\det g)\psi (\mathrm{Tr}(Y))⟨\lambda ,v⟩,$$

$$Z(v,s,\chi )={\int }_{{\mathrm{GL}}_n(\mathbb{R})}⟨\lambda ,\pi (\left(\begin{array}{cc}g& \\ & {\mathrm{I}}_n\end{array}\right))v⟩|\det g{|}^{s-\frac{1}{2}}\chi (\det g)dg$$

$${\text{Hom}}_H(V_{\pi },|\det {|}^{-s_0+\frac{1}{2}}{\chi }^{-1}(\det )\otimes |\det {|}^{s_0-\frac{1}{2}}(\chi \omega )(\det )),$$

$${\text{dim Hom}}_H(\pi ,|\det {|}^{-s_0+\frac{1}{2}}{\chi }^{-1}(\det )\otimes |\det {|}^{s_0-\frac{1}{2}}(\chi \omega )(\det ))\le 1.$$

$${\mathrm{H}}^{\ast }(\mathfrak{g},K^0{Z}^0,\pi \otimes F_{\nu }^{\vee })\ne 0,$$

$${\Lambda }_{s,\chi }\in {\mathrm{Hom}}_H(\pi ,|\det {|}^{-s+\frac{1}{2}}{\chi }^{-1}(\det )\otimes |\det {|}^{s-\frac{1}{2}}(\chi \omega )(\det )).$$

Full text

In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ for all complex values $s$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector using classical invariant theory, and hence proves the non-vanishing results of Sun in [24, Theorem 5.1] via a completely different method.

The research of Jiang is supported in part by the NSF Grants DMS–1600685 and DMS–1901802; that of Lin is supported in part by the China Scholarship Council No.201706245006; and that of Tian is is supported in part by AcRF Tier 1 grant R-146-000-277-114 of National University of Singapore.

1. 1 Introduction

2. 1.1 Cohomological representations of ${\mathrm{GL}}_{2n}({\mathbb{R}})$

3. 1.2 Shalika model and linear model

4. 1.3 Cohomological test vectors and non-vanishing property

5. 2 Cohomological Representations and Shalika Models

6. 3 Cohomological Representations and Linear Models

7. 3.1 The ${\mathrm{GL}}_{2}(\mathbb{R})$ case

8. 3.2 A new construction of linear model

9. 4 Cohomological Vectors in the Induced Representation

10. 4.1 Some reductions

11. 4.2 On certain minimal $K$-type functions

12. 4.3 On $K\cap H$-equivariant cohomological test vectors

13. 5 Non-vanishing of Archimedean Local Integrals

1. Introduction

Let $k$ be a number field, and ${\mathbb{A}}$ be the ring of adeles of $k$. Let $\pi$ be an irreducible cuspidal automorphic representation of the general linear group ${\mathrm{GL}}_{2n}({\mathbb{A}})$. The standard $L$-function $L(s,\pi\otimes\chi)$ of $\pi$, twisted by an idele-class character $\chi$ of $k^{\times}$, was first studied by R. Godement and H. Jacquet in 1972 ([8]), and then by the Rankin-Selberg convolution method of Jacquet, I. Piatetski-Shapiro and J. Shalika in 1983 ([13]). In 1993, S. Friedberg and Jacquet found in [7] a new global zeta integral for $L(s,\pi\otimes\chi)$, assuming that $\pi$ has a non-zero Shalika period.

Let $\omega=\omega_{\pi}$ be the central character of $\pi$ and take an idele-class character $\eta$ such that $\eta^{n}\cdot\omega=1$. The global zeta integral ${\mathcal{Z}}(\varphi_{\pi},\chi,\eta,s)$ of Friedberg-Jacquet is given by

[TABLE]

where $[{\mathrm{GL}}_{n}\times{\mathrm{GL}}_{n}]:=Z_{2n}({\mathbb{A}})({\mathrm{GL}}_{n}(k)\times{\mathrm{GL}}_{n}(k)){\backslash}({\mathrm{GL}}_{n}({\mathbb{A}})\times{\mathrm{GL}}_{n}({\mathbb{A}}))$, with $Z_{2n}$ the center of ${\mathrm{GL}}_{2n}$. In [7, Proposition 2.3], it is proved that ${\mathcal{Z}}(\varphi_{\pi},\chi,\eta,s)$ converges absolutely for all $s\in{\mathbb{C}}$ and for $\mathrm{Re}(s)$ sufficiently large, it is equal to the absolutely convergent integral

[TABLE]

where $V_{\varphi_{\pi}}$ is the global Shalika period of $\varphi_{\pi}$ that is defined as follows. Let $S$ be the Shalika subgroup of ${\mathrm{GL}}_{2n}$ consisting of matrices of the form

[TABLE]

where $x\in{\mathrm{Mat}}_{n}$ and $g\in{\mathrm{GL}}_{n}$. Define $\theta_{\eta}(s(x,g)):=\eta(\det g)\psi^{-1}({\mathrm{Tr}}(x))$ with a non-trivial additive character $\psi$ of $k{\backslash}{\mathbb{A}}$. The Shalika period of $\varphi_{\pi}$ is defined by

[TABLE]

By the local uniqueness of the Shalika model ([22], [4], and [5]), for the factorizable $\varphi_{\pi}=\otimes_{v}\varphi_{v}$, one has that $V_{\varphi_{\pi}}(h)=\prod_{v}V_{\varphi_{v}}(h_{v})$ with $V_{\varphi_{v}}(h_{v})$ being the local Shalika function associated to the local Shalika model at each place $v$, and an euler product decomposition:

[TABLE]

where the local zeta integrals are defined by

[TABLE]

Furthermore, it is proved that the local zeta integral ${\mathcal{Z}}_{v}(V_{\varphi_{v}},\chi_{v},s)$ is a holomorphic multiple of the local $L$-function $L(s,\pi_{v}\otimes\chi_{v})$ ([7] and [4]). It is clear that the Friedberg-Jacquet global zeta integral for $L(s,\pi\otimes\chi)$ is another natural generalization of the classical Hecke-type global zeta integral of Jacquet-Langlands for ${\mathrm{GL}}_{2}$ ([12]).

Among the three constructions of different global zeta integrals for $L(s,\pi\otimes\chi)$, it seems that the Friedberg-Jacquet global zeta integral for $L(s,\pi\otimes\chi)$ is better for applications with $\pi$ being cohomological, since the construction is closely related to the generalized modular symbols ([1] and [14]). We refer to [14] for more detailed discussions of the applications of the Friedberg-Jacquet integrals to the period relations of the critical values at different critical places for the automorphic $L$-functions $L(s,\pi\otimes\chi)$. For such important applications, it is technically very essential to establish the following properties related to the Friedberg-Jacquet integral for $L(s,\pi\otimes\chi)$:

1. (1)

Non-vanishing Hypothesis: The modular symbol at infinity is non-zero. 2. (2)

Uniform Cohomological Test Vector: The archimedean local zeta integral ${\mathcal{Z}}_{v}(V_{\varphi_{v}},\chi_{v},s)$ admits a uniform cohomological test vector $\varphi_{v}$ in the sense that

[TABLE]

holds as a meromorphic function of $s\in{\mathbb{C}}$.

For (1), B. Sun establishes in [24, Theorem 5.5] the non-vanishing hypothesis for real case by showing the existence of certain cohomological test vectors. For (2), the best result to the date is Sun’s existence of cohomological test vector in [24, Theorem 5.1], which shows that for any irreducible essentially tempered cohomological Casselman-Wallach representation $\pi_{v}$ of ${\mathrm{GL}}_{2n}(\mathbb{R})$ and every $s\in{\mathbb{C}}$, there exists a cohomological vector $\varphi_{v,s}$, depending on $s$, such that the normalized Friedberg-Jacquet integral

[TABLE]

As explained [14], this is not enough to obtain the global period relation of the critical values of the twisted standard $L$-functions $L(s,\pi\otimes\chi)$ at different critical places.

The objective of this paper is to develop a constructive approach towards Problems (1) and (2), which is complementary to the approach taken by Sun in [24]. In this paper, we do the real case, and leave the complex case to [18], which is similar, but has extra complications.

In the process of our understanding of the archimedean local zeta integrals of Friedberg-Jacquet, we find a new type of local integrals, which produce new linear functionals $\Lambda_{s,\chi}$ and give a different exppression of the linear models for ${\mathrm{GL}}_{2n}({\mathbb{R}})$. The explicitly constructed cohomological test vector, which is the most technical work of this paper (Section 4), turns out to be a uniform cohomological test vector that relates $\Lambda_{s,\chi}$ and the local twisted standard $L$-function $L(s,\pi\otimes\chi)$. With the known relation between the linear models, the Shalika models and the local zeta integrals of Friedberg-Jacquet, we deduce our main result (Theorem 1.2), which recovers the non-vanishing result of Sun in [24, Theorem 5.1], with explicitly constructed cohomological test vectors. Meanwhile, Theorem 1.2 shows that our explicitly constructed cohomological test vectors give a solution to Problem (2) on uniform cohomological test vector for the archimedean local zeta integral of Friedberg-Jacquet, up to an exponential type function in $s$. This exponential type function in $s$ will be removed with full details, including the complex case, in [14], which will lead to a complete proof of the global period relation of critical values of the twisted standard $L$-functions at different critical places for irreducible, regular algebraic, cuspidal automorphic representations of ${\mathrm{GL}}_{2n}$ of (generalized) symplectic type ([14]).

1.1. Cohomological representations of ${\mathrm{GL}}_{2n}({\mathbb{R}})$

Let $G={\mathrm{GL}}_{2n}(\mathbb{R})$, and $K={\mathrm{O}}_{2n}(\mathbb{R})$ be the maximal compact subgroup of $G$. Let $B$ be the Borel subgroup of $G$ consisting of all upper-triangular matrices in $G$. We fix the usual root system of $G$ so that $B$ contains all simple root vectors. Then the half sum of all positive roots, denoted by $\rho$, is

[TABLE]

It is clear that all standard parabolic subgroups of $G$ are in one-to-one correspondence with the ordered partition of $2n$. For instance, when $n=2$, we regard $4=1+3$ and $4=3+1$ as different partitions. Accordingly, they correspond to standard parabolic subgroups of ${\mathrm{GL}}_{4}({\mathbb{R}})$ whose Levi subgroups are ${\mathrm{GL}}_{1}({\mathbb{R}})\times{\mathrm{GL}}_{3}({\mathbb{R}})$ and ${\mathrm{GL}}_{3}({\mathbb{R}})\times{\mathrm{GL}}_{1}({\mathbb{R}})$ respectively.

Set

[TABLE]

Denoted by $Z$ the center of $G$. Let $d$ be the dimension of the quotient space ${\mathrm{Lie}}(H)/{\mathrm{Lie}}((K\cap H)Z)$, where ${\mathrm{Lie}}(H)$ is the Lie algebra of $H$. Then

[TABLE]

which, as suggested by [2] and exhibited in [9, Section 3.4], is the dimension of the modular symbol generated by the closed subgroup $H$. To fix notation, from now on, we will use capital letters $G,H$ etc. for certain Lie groups, $G^{0},H^{0}$ etc. for their identity components, German letters $\mathfrak{g},\mathfrak{h}$ etc. for their Lie algebras, and $\mathfrak{g}^{\mathbb{C}},\mathfrak{h}^{\mathbb{C}}$ for the complexifications of the Lie algebras.

Let $F_{\nu}$ be a highest weight representation of ${\mathrm{GL}}_{2n}(\mathbb{C})$ with highest weight $\nu$, which can be written as a vector of the following type:

[TABLE]

with $\nu_{1}\geq\nu_{2}\geq\cdots\geq\nu_{2n}$. The non-vanishing hypothesis involved in the above applications suggests that in this paper we only need to consider the irreducible essentially tempered Casselman-Wallach representations $(\pi,V_{\pi})$ of ${\mathrm{GL}}_{2n}(\mathbb{R})$ with property that the total relative Lie algebra cohomology

[TABLE]

is non-zero. By abuse of notation, we also use $\pi$ for its underlying $(\mathfrak{g},K)$-module when no confusion arises.

Now we recall the Langlands parameter for $\pi$. (For a general reference, see [6, Section 3]. Also see [19, Section 3.1] and [9, Section 3.4].) If the cohomology group

[TABLE]

is nonzero, then the highest weight $\nu$ satisfies the following purity condition:

[TABLE]

The integer $m$ is exactly the integer $w$ in [19, Section 3.1.4] and [9, Section 3.4]. We save the notation $w$ for Weyl elements. Moreover, we must also have that

[TABLE]

One should observe that the top non-vanishing degree $n^{2}+n-1$ is exactly the dimension of the modular symbol $d$ defined in (1.5). This is the key numerical coincidence, with which Sun is able to prove the non-vanishing of cohomological maps based on non-vanishing of one archimedean integral.

Given a highest weight $\nu$ and an integer $m$ satisfying (1.8), we define

[TABLE]

where $\rho$ is given in (1.4). If we write $\vec{l}=(l_{1},l_{2},\cdots,l_{2n})$, then

[TABLE]

We note that all $l_{i}$ share the same parity, which is different from the parity of $m$. For each positive integer $k$, we write $D_{k}$ for the relative discrete series of ${\mathrm{GL}}_{2}(\mathbb{R})$ with quadratic central character whose minimal $K$-type has highest weight $k+1$. Denote by $\eta_{k}$ the central character of $D_{k}$, which is given by

[TABLE]

The cohomological representation $\pi$ must be isomorphic to the normalized parabolically induced representation

[TABLE]

where $P$ is the standard parabolic subgroup of $G$ associated with the partition $[2^{n}]$. The central character $\omega_{\pi}$ of $\pi$ takes the form

[TABLE]

We define a character $\omega$ of $\mathbb{R}^{\times}$ as follows:

[TABLE]

Then $\omega$ is exactly the central character of $D_{l_{i}}\lvert\quad\rvert^{\frac{m}{2}}$, for all $i$ and $\omega_{\pi}(a{\mathrm{I}}_{2n})=(\omega(a))^{n}$. Clearly, the restriction of $\omega$ on $\{\pm 1\}$, denoted by $\omega_{0}$, is trivial when $m$ is even; and is the sign character when $m$ is odd. The highest weight of the minimal $K$-type of $\pi$, which is denoted by $\tau$, is $(l_{1}+1,l_{2}+1,\cdots,l_{n}+1)$. We will explain the highest weight for $K={\mathrm{O}}_{2n}(\mathbb{R})$ with more details in Subsection 4.2. The above discussion can be summarized in the following proposition.

Proposition 1.1**.**

Let $(\pi,V_{\pi})$ be an irreducible essentially tempered Casselman-Wallach representation of $G$ with property that

[TABLE]

Then $\pi$ is equivalent to the normalized induced representation

[TABLE]

as given in (1.12), and with the central character $\omega_{\pi}$ being given in (1.13). Moreover, the minimal $K$-type $\tau$ of $\pi$ has the highest weight $(l_{1}+1,l_{2}+1,\cdots,l_{n}+1)$.

1.2. Shalika model and linear model

Let $F=\mathbb{R}$ or $\mathbb{C}$. Let us fix a non-trivial unitary character $\psi$ of $F$ and a multiplicative character $\widetilde{\omega}$ of $F^{\times}$. For any positive integer $n^{\prime}$, we say a Casselman-Wallach representation $\sigma$ of ${\mathrm{GL}}_{2n^{\prime}}(F)$ has a non-zero $(\widetilde{\omega},\psi)-$Shalika model if there exists a non-zero continuous linear functional $\lambda$ on the Fréchet space $V_{\sigma}$, which is called a Shalika functional, such that

[TABLE]

for any $v\in V_{\sigma}$, $g\in{\mathrm{GL}}_{n^{\prime}}(F)$ and any $n^{\prime}\times n^{\prime}$ matrix $Y\in{\mathrm{Mat}}_{n^{\prime}}(F)$. In Section 2, we will show (in Theorem 2.1) that the normalized parabolic induction from two representations with non-zero $(\widetilde{\omega},\psi)-$Shalika models also admits a non-zero $(\widetilde{\omega},\psi)-$Shalika model. The existence of the $(\widetilde{\omega},\psi)-$Shalika model of the parabolic induction should be viewed as a direct archimedean analogue of [21, Theorem 1.1]. As a consequence, if we apply $F=\mathbb{R}$ and $\widetilde{\omega}=\omega$, the central character of each $D_{l_{j}}\lvert\quad\rvert^{\frac{m}{2}}$, we can conclude that the representation $\pi$ as in Proposition 1.1 also has a non-zero $(\omega,\psi)-$Shalika model. In the literature, the uniqueness of Shalika functional is proved in [4, Theorem 1.1] when $\widetilde{\omega}$ is trivial. For the general case, the same result is confirmed in [5, Theorem A]. For a character $\chi$ of $\mathbb{R}^{\times}$, the local archimedean integral of Friedberg-Jacquet at real place as in (1.3) can be re-written as

[TABLE]

for $v\in V_{\pi}$. We note that when both $\chi$ and $\omega_{\pi}$ are trivial, the integral in (1.16) is exactly the local integral considered in [2].

By [4, Theorem 3.1], the integral (1.16) converges absolutely when $\mathrm{Re}(s)$ is sufficiently large. $Z(v,s,\chi)$ is a homomorphic multiple of $L(s,\pi\otimes\chi)$ in the sense of meromorphic continuation and there exists a smooth vector $v\in V_{\pi}$ such that $Z(v,s,\chi)=L(s,\pi\otimes\chi)$. Thus whenever $s=s_{0}$ is not a pole of $L(s,\pi\otimes\chi)$, $Z(v,s,\chi)$ has no pole at $s=s_{0}$. This implies that the map: $v\mapsto Z(v,s_{0},\chi)$ defines a nonzero element in

[TABLE]

which is called the space of twisted linear functionals of $\pi$. The uniqueness of the twisted linear model is proved in [5]. In our scenario, we apply [5, Theorem B] and conclude that for all but countably many characters $\chi$,

[TABLE]

In fact, if $\lvert\det\rvert^{-s_{0}+\frac{1}{2}}\chi^{-1}(\det)\otimes\lvert\det\rvert^{s_{0}-\frac{1}{2}}(\chi\omega)(\det)$ is a good character of $H$, then (1.17) holds. For the precise definition of a good character of $H$, we refer to [5]. Finally we remark that when $\chi$ and $\omega$ are both trivial and $s_{0}=\frac{1}{2}$, the uniqueness theorem is proved in [3].

1.3. Cohomological test vectors and non-vanishing property

As explained above, any irreducible essentially tempered Casselman-Wallach representation $\pi$ of $G$ with non-zero cohomology must be of the form given in Proposition 1.1, and such a $\pi$ must have a non-zero Shalika model. Yet, even with an explicit vector $v\in V_{\pi}$, the Shalika function $\langle\lambda,\pi(\begin{pmatrix}g&\\ &{\mathrm{I}}_{n}\end{pmatrix})v\rangle$ is hard to evaluate, let alone the local Friedberg-Jacquet integral $Z(v,s,\chi)$. Alternatively, we construct explicitly another twisted linear functional $\Lambda_{s,\chi}$ given by an explicit integral in (3.10). The advantage of this newly constructed twisted linear functional is that the value of $\Lambda_{s,\chi}$ at a vector $f\in V_{\pi}$ with a certain right $K\cap H$-equivariance (realize $f$ as a function using parabolic induction) is easy to compute once we know the value of $f$ at one certain Weyl element $w$ (see (3.12)). We recall that when $\pi$ is cohomological, a cohomological vector $v$ of $\pi$ is defined to be a vector $v$ belonging to the minimal $K$-type $\tau$ of $\pi$. Thus, to prove the non-vanishing result of Friedberg-Jacquet integral $Z(v,s,\chi)$, it suffices to construct explicitly a function $f\in V_{\pi}$ in the minimal $K$-type $\tau$ with a certain right $K\cap H$-equivariance, which is a highly non-trivial task.

Let us only outline the key ingredients in the explicit construction of the cohomological vector $f\in V_{\pi}$.

1. (1)

With some reductions discussed in Subsection 4.1 via the compact induction model, it suffices to construct a ${\mathbb{C}}$-valued function, which will be denoted by $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$, in the minimal $K$-type of a certain induced representation $\pi_{sc}$ defined in (4.8). 2. (2)

According to the representation theory of compact Lie groups, we have the fact that under the left $K$-action, if a function $\varphi$ is a lowest or highest weight vector of an irreducible $K$-submodule $\alpha$ of $C^{\infty}(K)$, then it generates $\alpha^{*}$ under the right $K$-action. See Lemma 4.5. 3. (3)

Via Lemma 4.5 and the classical invariant theory, we can construct a weight-building function $F_{wt,\vec{N}}(x)\in C^{\infty}(K)$ in Corollary 4.9, which will help us build up the weight of $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$. Two determinant functions $F_{\mathrm{left}}(x),F_{\mathrm{right}}(x)$ that govern the right $(K\cap H)$-equivariance are constructed in Proposition 4.10. Then define $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ to be a product of them (see Theorem 4.12).

The explicit construction of the cohomological vector $v=f(k)$ as in Corollary 4.13 based on $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}$ leads to the main theorem of this paper:

Theorem 1.2** (Main Theorem).**

Let $(\pi,V_{\pi})$ be an irreducible essentially tempered Casselman-Wallach representation of $G$ with property that

[TABLE]

and $\Lambda_{s,\chi}$ be the linear functional on $V_{\pi}$ defined in (3.10). Then the following hold.

1. (1)

Whenever $s$ is not a pole of the $L$-function $L(s,\pi\otimes\chi)$,

[TABLE] 2. (2)

There exists a cohomological vector $v=f(k)$ as explicitly constructed in Corollary 4.13 such that as a meromorphic function of $s\in{\mathbb{C}},$

[TABLE]

Remark 1.3**.**

It turns out that the integral $\Lambda_{s,\chi}$ constructed in this paper is exactly the Friedberg-Jacquet integral, up to suitable normalization of Haar measures. The proof will be given in [14] with full details, together with an application to the global period relations of critical values of the twisted standard $L$-functions at different critical places.

The most technical parts of the paper are: one is to construct the new linear functional $\Lambda_{s,\chi}$ in Section 3.2; and the other is the explicit construction of the right cohomological vector $v=f(k)$ of $\pi$ belonging to the minimal $K$-type and having the desired property in Section 4.

Finally, we would like to thank Lei Zhang for helpful conversation during our preparation of this paper, and thank Binyong Sun for sending us his preprints [5] and [24] when we were writing up this paper. We would also like to thank the referee for very helpful comments and suggestions.

2. Cohomological Representations and Shalika Models

The goal of this Section is to prove a hereditary property of Shalika models with respect to normalized parabolic induction. Let us first start from the case of ${\mathrm{GL}}_{2}.$ For future applications, we will consider both the real case and complex case in this Section, i.e. $F=\mathbb{R}$ or $\mathbb{C}$.

Suppose that we have $l$ Frechét spaces $V_{1},V_{2},\cdots,V_{l}$. Let $\lambda_{j}$ be a continuous linear functional on $V_{j}$ ($j=1,2,\cdots,l$). Then $\bigotimes_{j=1}^{l}\lambda_{j}$ is a continuous linear functional on the projective tensor space $\widehat{\bigotimes}_{j=1}^{l}V_{j}$, which is also a Fréchet space.

Let $\sigma$ be a generic Casselman-Wallach representation of ${\mathrm{GL}}_{2}(F)$ with a central character $\omega_{\sigma}$. We fix a nontrivial unitary character $\psi$ of $F$. Then $\sigma$ admits a non-zero Whittaker model $\mathcal{W}(\sigma,\psi)$, i.e. there exists a continuous linear functional $\lambda_{\sigma}$ on the Fréchet space $V_{\sigma}$ such that

[TABLE]

Thus

[TABLE]

which exactly coincides with (1.15) for $n=1$. Hence any such $\sigma$ has a non-zero $(\omega_{\sigma},\psi)-$Shalika model.

The main theorem in this Section is formulated below, which can be regarded as an archimedean analogue of [21, Theorem 1.1].

Theorem 2.1**.**

Let $F=\mathbb{R}$ or $\mathbb{C}$ be an archimedean local field. Let $\omega$ be a character of $F^{\times}$ and $\psi$ be a nontrivial unitary character of $F$. For two even positive integers $n_{1}=2m_{1}$ and $n_{2}=2m_{2}$, take two Casselman-Wallach representations $\pi_{1}$ and $\pi_{2}$ of ${\mathrm{GL}}_{n_{1}}(F)$ and ${\mathrm{GL}}_{n_{2}}(F)$, respectively, and assume that both $\pi_{1}$ and $\pi_{2}$ have $(\omega,\psi)-$Shalika models. Then the normalized parabolic induction $\pi:={\mathrm{Ind}}_{P_{n_{1},n_{2}}(F)}^{{\mathrm{GL}}_{n_{1}+n_{2}}(F)}\pi_{1}\otimes\pi_{2}$ also has a non-zero $(\omega,\psi)-$Shalika model. Here $P_{n_{1},n_{2}}$ is a standard parabolic subgroup of ${\mathrm{GL}}_{n_{1}+n_{2}}$ with its Levi part isomorphic to ${\mathrm{GL}}_{n_{1}}\times{\mathrm{GL}}_{n_{2}}$.

The proof of Theorem 2.1 will occupy the rest of this Section. Our proof is very similar to that of [21, Theorem 1.1], and we borrow some continuity arguments from [14, Section 6.3]. We start with some estimates on Shalika functionals.

Lemma 2.2**.**

Let $k$ be a positive integer and $\sigma$ be a Casselman-Wallach representation of ${\mathrm{GL}}_{2k}(F)$ with a Shalika functional $\lambda_{\sigma}$. Then there exists a positive integer $M_{0}$ with property that for any integer $M\geq M_{0}$ and any polynomial $P$ on ${\mathrm{Mat}}_{k}(F)$, there exists a continuous seminorm $\beta_{P,M}$, such that the following estimate

[TABLE]

holds for all $g\in{\mathrm{GL}}_{k}(F)$ and $v\in V_{\sigma}$.

Proof.

In [4], the authors consider the Shalika functional when $\omega$ (in (1.15)) is trivial. Their proof of [4, Lemma 3.4] is still valid when $\omega$ is non-trivial. Also, we note that in the proof of [4, Lemma 3.4], the authors define a norm $\lVert\cdot\rVert$ on ${\mathrm{GL}}_{2k}(F)$ and use the fact that the product $\lVert\begin{pmatrix}g&\\ &{\mathrm{I}}_{k}\end{pmatrix}\rVert^{M_{0}}\cdot\lvert\det{g}\rvert^{M_{0}}$ is a polynomial. Hence for any integer $M\geq M_{0}$, $\lVert\begin{pmatrix}g&\\ &{\mathrm{I}}_{k}\end{pmatrix}\rVert^{M_{0}}\cdot\lvert\det{g}\rvert^{M}$ is still a polynomial. Then a word-by-word repetition of the proof of [4, Lemma 3.4] will confirm the lemma here. We omit the details here. ∎

Corollary 2.3**.**

With $k$, $\sigma$, and $\lambda_{\sigma}$ as given in Lemma 2.2, there exists an positive integer $M_{0}$ and a real number $m_{0}$ with property that for any positive integer $M\geq M_{0}$, there exists a continuous seminorm $\beta_{M}$ satisfying

[TABLE]

for all $g,h\in{\mathrm{GL}}_{k}(F)$, and $v\in V_{\sigma}$.

Proof.

Let $\omega=\widetilde{\omega}$ be the character associated with the given Shalika functional $\lambda_{\sigma}$ as in (1.15). Then

[TABLE]

We deduce Corollary 2.3 by writing $\lvert\omega(\det h)\rvert=\lvert\det h\rvert^{m_{0}}$ and applying Lemma 2.2 to the case of $P=1$. ∎

Now we apply such general estimates to the case of Theorem 2.1 and obtain the following estimate.

Corollary 2.4**.**

Take $n_{1}=2m_{1}$, $n_{2}=2m_{2}$, $\pi_{1}$, $\pi_{2}$, and $\pi$ as in Theorem 2.1. Write $m=m_{1}+m_{2}$ and denote by $\lambda_{1},\lambda_{2}$ the Shalika functionals on $V_{\pi_{1}},V_{\pi_{2}}$. Write $K_{m}$ for the maximal compact subgroup of ${\mathrm{GL}}_{m}(F)$. There exists a positive integer $M_{0}$ and a real number $m_{0}$ such that for any positive integer $M\geq M_{0}$, the following property holds: there exists a continuous seminorm of $q_{M}$ on the Fréchet space $V_{\pi}$, depending on $M$, such that for all $f\in V_{\pi}$, the following estimate

[TABLE]

holds for any $a\in{\mathrm{GL}}_{m_{1}}(F)$, $b\in{\mathrm{GL}}_{m_{2}}(F)$ and $k\in K_{m}$.

Proof.

By Corollary 2.3, there exists a positive integer $M_{0}$ with the property that for any positive integer $M\geq M_{0}$, there exists a continuous seminorm $q_{M}^{\prime}$ on $V_{\pi_{1}}\widehat{\otimes}V_{\pi_{2}}$ such that the following estimate holds for all $v\in V_{\pi_{1}}\widehat{\otimes}V_{\pi_{2}}$:

[TABLE]

Define

[TABLE]

where $K_{n}$ is the standard maximal compact subgroup of ${\mathrm{GL}}_{n}(F).$ We refer the reader to [28, 10.1.1] for the Fréchet topology of the parabolically induced representation $V_{\pi}$. Under such a Fréchet topology, the function $q_{M}$ defined in (2.7) is a continuous seminorm on $V_{\pi}.$ Thus, the estimate (2.4) follows directly from (2.5). We are done. ∎

We also need a lemma on the diagonal matrix appearing in the Iwasawa decomposition of a lower unipotent matrix.

Lemma 2.5**.**

Let $m_{1},m_{2}$ be two positive integers and set $m=m_{1}+m_{2}.$ Let $\overline{u}(x):=\begin{pmatrix}{\mathrm{I}}_{m_{1}}&\\ x&{\mathrm{I}}_{m_{2}}\end{pmatrix}$ be an element in the unipotent radical of the lower parabolic subgroup of ${\mathrm{GL}}_{m}(F)$, and

[TABLE]

be its Iwasawa decomposition, where $t(x)={\mathrm{diag}}(t_{1}(x),t_{2}(x),\cdots,t_{m}(x))$, with all $t_{i}(x)>0$. Here $u(x)$ lives in the standard maximal unipotent subgroup, $k(x)$ lives in the maximal compact subgroup. Define also

[TABLE]

Write x=\left(\begin{array}[]{c}x_{1}\\ x_{2}\\ \vdots\\ x_{m_{2}}\\ \end{array}\right), where each $x_{j}$ is a row vector in $F^{m_{1}}$. Denote by $\lVert\cdot\rVert$ the norm induced by the standard inner product in $F^{m_{1}}$. Then the following estimate holds

[TABLE]

Proof.

The following proof is a standard argument using the exterior algebra. Let $V:=F^{m}$ be the vector space equipped with the standard inner product. We also write $\lVert\cdot\rVert$ for the norm induced by this inner product when there is no confusion on the dimension of the vector space. Let $e_{1}$, $e_{2},\cdots,e_{m}$ be the standard orthonormal basis of $V$. Then every vector in $V$ is regarded as a row vector. For any positive integer $p$, the vector space $\bigwedge^{p}V$ of the $p$-th exterior power of $V$ is equipped with a standard inner product $\langle\cdot,\cdot\rangle_{p}$ defined by

[TABLE]

Denote by $\lVert\cdot\rVert_{p}$ the norm induced by the inner product $\langle\cdot,\cdot\rangle_{p}$. Then

[TABLE]

Under the inner product $\langle\cdot,\cdot\rangle_{p}$, all $e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{p}}$ $(1\leq i_{1}<i_{2}<\cdots<i_{p}\leq n)$ form an orthonormal basis of $\bigwedge^{p}V$. The group ${\mathrm{GL}}_{m}(F)$ acts on $V$ by right multiplication, and then induces an action on $\bigwedge^{p}V$. Under such an action, the maximal compact subgroup preserves the inner product $\langle\cdot,\cdot\rangle_{p}$. We also note that for any $1\leq i\leq m$ and any $u$ in the standard maximal unipotent subgroup, one has

[TABLE]

Thus, using the Iwasawa decomposition $\overline{u}(x)=u(x)t(x)k(x)$, we obtain that

[TABLE]

We also set $x_{j}^{\prime}=(x_{j},0)\in F^{m}$ for $j=1,2\cdots,m_{2}$. Then for each $m_{1}<i\leq m$,

[TABLE]

In particular, when $i=m_{1}+1$, we combine (2.10), (2.11) and (2), and obtain that

[TABLE]

We also have

[TABLE]

Thus, when $i=m_{1}+1$, we find

[TABLE]

This completes the proof of the Lemma. ∎

Proof of Theorem 2.1.

To simplify our notation in this proof, we set that $n=n_{1}+n_{2}$ and $m=m_{1}+m_{2}$. (These $n$ and $m$ have different meanings from those in the Introduction.) Let $\lambda_{1}$ and $\lambda_{2}$ be the Shalika functionals of $\pi_{1}$ and $\pi_{2}$ respectively. For each positive integer $k$, we write $I_{k}$ for the $k\times k$ identity matrix. We consider the Weyl element

[TABLE]

Take any function $f\in V_{\pi}$, we consider the function $\Phi(g;f)$ on ${\mathrm{GL}}_{n}$ defined by

[TABLE]

Then $\Phi(g;f)=\Phi({\mathrm{I}}_{n};\pi(g)f)$. Arguing as [21, Lemma 3.1], we can easily show that

[TABLE]

and

[TABLE]

Now we consider the integral

[TABLE]

Then $H(g;f)=H({\mathrm{I}}_{n};\pi(g)f).$ As [21, Lemma 3.2], we aim to show the integral (2.18) converges absolutely. We also need to prove that the linear map $f\mapsto H({\mathrm{I}}_{n};f)$ is continuous under the Fréchet topology of $V_{\pi}$. We write the integral (2.18) as

[TABLE]

Replacing $f$ by $\pi(w^{-1}g)f$, we only need to justify the absolute convergence and the continuity property of

[TABLE]

We define $\overline{u}(x)$ to be the $m\times m$-matrix

[TABLE]

Let $\overline{u}(x)=u(x)t(x)k(x)$ be the Iwasawa decomposition of $\overline{u}(x)$ in ${\mathrm{GL}}_{m}(F)$ as in Lemma 2.5. We also define $t_{i}(x)$, $a_{1}(x)$, $a_{2}(x)$ as in Lemma 2.5. We further decompose $u(x)$ as

[TABLE]

where $u_{1}(x)$ and $u_{2}(x)$ lives in the standard maximal unipotent subgroup of ${\mathrm{GL}}_{m_{1}}(F)$ and ${\mathrm{GL}}_{m_{2}}(F)$ respectively, and $u_{3}(x)$ lives in the standard unipotent radical of the parabolic subgroup of ${\mathrm{GL}}_{m}(F)$ of type $(m_{1},m_{2})$. It follows (using the notation introduced in the proof of Corollary 2.4) that

[TABLE]

where $\delta(a_{1}(x),a_{2}(x))$ is the modular character arising from equivariance for the normalized parabolic induction. More precisely,

[TABLE]

By Corollary 2.4, there exists a positive integer $M_{0}$ and a real number $m_{0}$ with the property that for any positive integer $M\geq M_{0}$, there exists a continuous seminorm $q_{M}$ on $V_{\pi}$, the following estimate holds:

[TABLE]

Note that $\det(a_{1}(x))\cdot\det(a_{2}(x))=1$. Thus, combining (2.22) and (2.24), and by the estimate (2.9) in Lemma 2.5, we can choose a sufficiently large positive integer $M$ such that

[TABLE]

This implies that the integral (2.19) converges absolutely and defines a continuous linear functional on $V_{\pi}$. Thus, the integral (2.18) converges absolutely and the linear functional $f\mapsto H({\mathrm{I}}_{n};f)$ is continuous.

According to (2.16), we can see that

[TABLE]

Denote by $V^{-}$ the lower unipotent subgroup consisting all matrices of the form

[TABLE]

Let $P:=P_{m_{1},m_{2}}$ be the standard parabolic subgroup of ${\mathrm{GL}}_{m}(F)$ associated with the partition $m=m_{1}+m_{2}$. The group ${\mathrm{GL}}_{m}(F)$ diagonally embeds into ${\mathrm{GL}}_{n}(F)$ with image ${\mathrm{GL}}_{m}^{\triangle}(F)$. For any subgroup $L$ of ${\mathrm{GL}}_{m}(F)$ and any matrix $x\in L$, we also write $L^{\triangle}$ and $x^{\triangle}$ for the image of the embedding

[TABLE]

Let $p=\begin{pmatrix}g_{1}&x\\ &g_{2}\end{pmatrix}\in P$. Then by (2.17), it is easy to see that $H(g;f)$ also satisfies the following equivariant property:

[TABLE]

Thus, we can find a test function $\varphi(h^{\triangle};f)\in C^{\infty}_{c}({\mathrm{GL}}_{m}^{\triangle}(F))$ such that

[TABLE]

where $d_{l}p$ is the left Haar measure on $P$. Now we define a functional $S(f)$ on $V_{\pi}$ as follows:

[TABLE]

We rewrite (2.29) as

[TABLE]

It is clear that $S(f)$ is well defined, i.e. the integral in (2.29) is convergent and independent on the choice of $\varphi(h^{\triangle};f)$. Thus combining (2.29), (2.30) and (2.26), we see

[TABLE]

Thus, the linear map $f\mapsto S(f)$ satisfies the desired equivariant property of a Shalika functional. Next, we show that this linear map is non-zero.

To show this non-vanishing property, we use the Bruhat decomposition of ${\mathrm{GL}}_{m}(F)$ in the integral (2.29). Let $N^{-}$ be the unipotent radical of the lower parabolic subgroup associated to the partition $m=m_{1}+m_{2}$. Then we have that for all $f\in V_{\pi}$,

[TABLE]

Note that $V^{-}\cdot w^{-1}N^{-}w$ is a subgroup of the unipotent radical $U^{-}$ of the lower parabolic subgroup of ${\mathrm{GL}}_{n}(F)$ associated to the partition $n=n_{1}+n_{2}$. We choose a positive test function $\varphi(g)\in C^{\infty}_{c}({\mathrm{GL}}_{n}(F))$ whose restriction to the subgroup $V^{-}\cdot w^{-1}N^{-}w$ is positive on a subset with positive measure. Since $\lambda_{1}$, $\lambda_{2}$ are non-zero Shalika functionals, we can find two vectors $v_{1},v_{2}$ in $V_{\pi_{1}}$ and $V_{\pi_{2}}$ resp, such that

[TABLE]

We define a function $\widetilde{f}$ on $U^{-}$ by $\widetilde{f}(u^{-})=\varphi(u^{-})v_{1}\otimes v_{2}$, and extend it to a function on ${\mathrm{GL}}_{n}(F)$ by the equivariant property of the induced representation $\pi$. Then the function

[TABLE]

satisfies

[TABLE]

Thus the map $f\mapsto S(f)$ is nonzero.

We finally check that the linear map $f\mapsto S(f)$ is continuous under the Fréchet topology of the induced representation. By (2.30),

[TABLE]

For any fixed $k\in K_{m}$, the linear functional $f\mapsto H({\mathrm{I}}_{n};\pi(k^{\triangle})f)\omega^{-1}(\det k)$ is continuous. It follows that with the fixed $f\in V_{\pi}$, the function $k\mapsto H({\mathrm{I}}_{n};\pi(k^{\triangle})f)\omega^{-1}(\det k)$ defined on $K_{m}$ is bounded, since $K_{m}$ is compact. Thus, by the Uniform Boundedness Principle, the family of continuous linear functionals $f\mapsto H({\mathrm{I}}_{n};\pi(k^{\triangle})f)\omega^{-1}(\det k)$ indexed by $k\in K_{m}$ is equicontinuous. Thus,

[TABLE]

is a continuous linear functional on $V_{\pi}$.

∎

Corollary 2.6**.**

Any irreducible essentially tempered Casselman-Wallach representation $(\pi,V_{\pi})$ of $G={\mathrm{GL}}_{2n}(\mathbb{R})$ with

[TABLE]

has a non-zero Shalika model defined by the Shalika functional as in (1.15).

It is clear that Corollary 2.6 follows from Theorem 2.1 and the example on ${\mathrm{GL}}_{2}(F)$ discussed at the beginning of this section.

3. Cohomological Representations and Linear Models

According to Corollary 2.6, the cohomological representation $\pi$ of $G={\mathrm{GL}}_{2n}({\mathbb{R}})$ as in Theorem 1.2 has a non-zero Shalika model, and hence the archimedean local integral $Z(v,s,\chi)$ of Friedberg-Jacquet as defined in (1.16) is nonzero. However, due to a lack of nice formula for the Shalika functional, it is not easy to directly construct a cohomological test vector for the Friedberg-Jacquet integral $Z(v,s,\chi)$. Instead, we construct in this Section an explicit linear model $\Lambda_{s,\chi}$, with which it is easier to obtain a cohomological test vector, as discussed in the next Section.

3.1. The ${\mathrm{GL}}_{2}(\mathbb{R})$ case

Let us first review the work of A. Popa on the ${\mathrm{GL}}_{2}(\mathbb{R})$ case. We will only recall his results on discrete series, which will be used in our further computation. For detailed discussion on the principal series of ${\mathrm{GL}}_{2}(\mathbb{R})$ and of ${\mathrm{GL}}_{2}(\mathbb{C})$, we refer to his paper [23]. His work on ${\mathrm{GL}}_{2}(\mathbb{C})$ is also very helpful in the complex case that will be considered in [18].

In this Section, we fix a non-trivial unitary character $\psi$ of $\mathbb{R}$ and a character $\chi$ of $\mathbb{R}^{\times}$. Let $k$ be a positive integer and $D_{k}$ be the relative discrete series (which is assumed to be a Casselman-Wallach representation) in the Introduction. The minimal $K$-type of $D_{k}$ (whose central character is $\eta_{k}$) is denoted by $\tau_{k}$. Then $\tau_{k}$ is a two dimensional irreducible representation of ${\mathrm{O}}_{2}(\mathbb{R})$. We can take basis $v_{k}$ and $v_{-k}$ of $\tau_{k}$ such that $v_{k}$ and $v_{-k}$ are weight vectors with weight $\pm(k+1)$ under the action of ${\mathrm{SO}}_{2}(\mathbb{R})$. Moreover, $v_{k}$ and $v_{-k}$ are related by the action of $\epsilon=\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}$, i.e. $D_{k}(\epsilon)v_{k}=v_{-k}$. The two dimensional space $\tau_{k}$ contains a one-dimensional subspace denoted by $\tau_{k}^{\epsilon,\chi}$ (in [23], the notation $\mathcal{W}^{T}$ was used for this invariant subspace), which is spanned by the vector $v_{k}+\chi(-1)v_{-k}$. Note that

[TABLE]

Given an irreducible generic Casselman-Wallach representation $\sigma$ of ${\mathrm{GL}}_{2}(\mathbb{R})$ with a Whittaker model $\mathcal{W}(\sigma,\psi)$, it is clear ([12]) that the archimedean Hecke integral

[TABLE]

has a meromorphic continuation to the whole complex plane. It is a holomorphic multiple of the $L$-function $L(s,\sigma\otimes\chi)$. Whenever $s=s_{0}$ is not a pole of the $L$-function $L(s,\sigma\otimes\chi)$, $\lambda_{s_{0},\chi,\sigma}$ defines a continuous linear functional on $\sigma$. Now if apply to the case of $\sigma=D_{k}$, then whenever $s=s_{0}$ is not a pole, $\lambda_{s_{0},\chi,{D_{k}}}$ defines a nonzero element in

[TABLE]

The non-vanishing property of $\lambda_{s_{0},\chi,{D_{k}}}$ can be deduced from the following result of Popa (we will rephrase it using our notation):

Proposition 3.1**.**

[23, Theorem 1]** There exists a vector $v\in\tau_{k}^{\epsilon,\chi}$ such that when $\mathrm{Re}(s)$ is sufficiently large,

[TABLE]

Since $\tau_{k}^{\epsilon,\chi}$ is one-dimensional, we can normalize the basis so that

[TABLE]

The same result also holds once we twist a determinant character on the relative discrete series.

Corollary 3.2**.**

Given $\alpha\in\mathbb{C}$, let $\sigma_{k}:=D_{k}\otimes\lvert\cdot\rvert^{\alpha}$ and $\lambda_{s,\chi,{\sigma_{k}}}$ be the continuous linear functional defined in (3.1). Then for every $v\in V_{\sigma_{k}}$, $\lambda_{s,\chi,{\sigma_{k}}}(v)$ is a holomorphic multiple of $L(s,\sigma_{k}\otimes\chi)$. Whenever $s=s_{0}$ is not a pole of $L(s,\sigma_{k}\otimes\chi)$, $\lambda_{s_{0},\chi,{\sigma_{k}}}(v)$ defines a nonzero element in

[TABLE]

where $\widetilde{\eta_{k}}=\eta_{k}\lvert\cdot\rvert^{\alpha}$ is the central character of $\sigma_{k}$. Moreover,

[TABLE]

3.2. A new construction of linear model

In this Subsection, we retain the notation in the Introduction and assume that $n\geq 2$. Let $\pi$ be the parabolically induced representation in (1.12). In Section 2, we have shown that $\pi$ has a Shalika model (Corollary 2.6). Thus the local integral (1.16) defines a twisted linear model of $\pi$, i.e. (1.16) defines a non-zero element in

[TABLE]

whenever $s$ is not a pole of the $L$-function, as we explained in the Introduction. The goal of this Subsection is to construct another linear model without using the Shalika model.

The basic idea comes from standard Bruhat theory. We first define a Weyl element $w$. Let $e_{1},e_{2},\cdots,e_{2n}$ be the standard basis of the vector space ${\mathbb{R}}^{2n}$, where ${\mathbb{R}}^{2n}$ is realized as a vector space of column vectors. We consider the following Weyl element

[TABLE]

Denote by $P$ the standard parabolic subgroup of $G$ corresponding to the partition $[2^{n}]$ of $2n$, as before. We look at the homogeneous space $P\backslash G$ and let $H$ acts on $P\backslash G$ by right translation. It was shown in [20, Proposition 3.4] that there are finitely many orbits. In this paper, it is sufficient for us to consider the orbit $P\backslash PwH$. By a direct matrix computation, it is easy to show that the stabilizer is

[TABLE]

where $B_{1}\times B_{2}$ is the standard upper Borel subgroup of $H={\mathrm{GL}}_{n}({\mathbb{R}})\times{\mathrm{GL}}_{n}({\mathbb{R}}).$ Thus the orbit $P\backslash PwH$ is homeomorphic to $(B_{1}\times B_{2})\backslash H$. In particular, the orbit $P\backslash PwH$ is closed. In the following, we are going to construct an integral on this closed orbit which represents a linear functional in

[TABLE]

We still fix a character $\chi$ of $\mathbb{R}^{\times}$ as we did in Subsection 3.1. Let $P=MU$ be the Levi decomposition of $P$. Define $H^{\prime}=\mathrm{Ad}(w)H$. Then $H^{\prime}$ is also isomorphic to ${\mathrm{GL}}_{n}(\mathbb{R})\times{\mathrm{GL}}_{n}(\mathbb{R})$. Let $N=N_{1}\times N_{2}$ be the unipotent radical of $B_{1}\times B_{2}$. The standard maximal unipotent subgroup $N$ of $H$ is mapped onto the standard maximal unipotent subgroup $N^{\prime}$ of $H^{\prime}$. We notice that $N^{\prime}$ is a subgroup of $U$.

Take $\varphi\in V_{\pi}$. Then $\varphi$ is a smooth function on $G$ with value in $\widehat{\bigotimes}_{i=1}^{n}V_{\sigma_{i}}$, where $\sigma_{i}:=D_{l_{i}}\lvert\quad\rvert^{\frac{m}{2}}$. As we explained in the Introduction, all $l_{i}$ have the same parity, and hence all $\sigma_{i}$ have the same central character $\omega$ (defined in (1.14)). For each $i$, whenever $s=s_{0}$ is not a pole of $L(s,\sigma_{i}\otimes\chi)$, we denote by $\lambda_{s_{0},i}$ the non-zero element in

[TABLE]

as in Corollary 3.2.

To simplify notation, we define two characters of ${\mathbb{R}}^{\times}$:

[TABLE]

Let us consider a function on $G$ defined by

[TABLE]

Then $F_{s}(g;\varphi)=F_{s}({\mathrm{I}}_{2n};\pi(g)\varphi)$. Since $N^{\prime}$ is a subgroup of $U$, it is easy to check that for $n_{1}\in N_{1}$, $n_{2}\in N_{2}$,

[TABLE]

By the equivariance of $\lambda_{s,i}$, it is also easy to check the equivariance of $F(g;\varphi)$ on the torus: for all $n\times n$ diagonal invertible matrices $a_{1},a_{2}$, we have that

[TABLE]

where the modular character can be explicitly computed as follows

[TABLE]

It follows that $F_{s}(g)$ satisfies a $B_{1}\times B_{2}-$equivariant property: for any $(b_{1},b_{2})\in B_{1}\times B_{2},$

[TABLE]

Thus the following convergent integral defines a continuous linear functional $\Lambda_{s,\chi}$ on $V_{\pi}$

[TABLE]

It is easy to see that $\Lambda_{s,\chi}\in\textrm{Hom}_{H}(\pi,\chi_{1,s}(\det)\otimes\chi_{2,s}(\det))$.

In terms of $\varphi$, $\Lambda_{s,\chi}(\varphi)$ can be written as:

[TABLE]

where $\widetilde{\varphi}$ is obtained by averaging $\varphi$ against $\chi_{1,s}^{-1}(\det k_{1})\chi_{2,s}^{-1}(\det k_{2})$ over the compact group $K\cap H$. In particular, if $\varphi$ satisfies the right $K\cap H$-equivariant property:

[TABLE]

where $\chi_{0}$ ($\omega_{0}$ resp.) is the restriction of the character $\chi$ ($\omega$ resp.) on $\{\pm 1\}$, then

[TABLE]

The following Proposition gives the desired property of $\Lambda_{s,\chi}$.

Proposition 3.3**.**

For every $\varphi\in V_{\pi}$, $\Lambda_{s,\chi}(\varphi)$ defined by (3.10) has a meromorphic continuation in $s$ to the whole complex plane. It is a holomorphic multiple of $L(s,\pi\otimes\chi)$ and defines an element in the space

[TABLE]

which is the same as ${\mathrm{Hom}}_{H}(\pi,\lvert\det\rvert^{-s+\frac{1}{2}}\chi^{-1}(\det)\otimes\lvert\det\rvert^{s-\frac{1}{2}}(\chi\omega)(\det))$, whenever $s$ is not a pole of the $L$-function.

Proof.

This is a direct consequence of Corollary 3.2. ∎

We close this Section by the following remark:

Remark 3.4**.**

One should be able to prove that $\Lambda_{s,\chi}$ is a non-zero linear functional by imitating the proof of Theorem 2.1. Since later, we will prove a sharper result that $\Lambda_{s,\chi}$ does not vanish on the minimal $K$-type $\tau$ of $\pi$, we will not discuss the non-vanishing property here.

4. Cohomological Vectors in the Induced Representation

The goal of this section is to explicitly construct a cohomological vector of $\pi$ with the desired non-vanishing property for Theorem 1.2. We retain all notation in the Introduction. Since $\pi$ (given in (1.12)) is realized as a parabolically induced representation, every function $f\in V_{\pi}$ is determined by its value on the maximal compact subgroup $K$. As (3.12) suggests, we only care about the value of cohomological function at the Weyl element $w\in K$ as defined in (3.4). Thus, it is convenient to work with the compact induction model of $\pi$. Let us start with some reductions and outline our strategy of the construction of the function in the minimal $K$-type.

4.1. Some reductions

Let $\pi$ be the parabolically induced representation in (1.12), where $P$ is the standard parabolic subgroup of $G$ corresponding to the partition $[2^{n}]$ with Levi decomposition $P=MU$. Then $M\cap K$ is just a product of $n$ copies of ${\mathrm{O}}_{2}(\mathbb{R})$. We write a general element in $(M\cap K)^{0}$ as

[TABLE]

For the component group of $M\cap K$, we write

[TABLE]

where all $\epsilon_{i}\in\{\pm 1\}$. For simplicity, for any representation $\rho_{0}$ of ${\mathrm{SO}}_{2}(\mathbb{R})$, we write $\rho_{0}(\theta)$ for $\rho_{0}(\begin{pmatrix}\cos{\theta}&\sin{\theta}\\ -\sin{\theta}&\cos{\theta}\end{pmatrix})$. We denote by $\chi_{l}$, for each integer $l$, the character of ${\mathrm{SO}}_{2}(\mathbb{R})$ that sends $\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}$ to $e^{i\cdot l\theta}$.

Set $\sigma_{j}:=D_{l_{j}}\lvert\quad\rvert^{\frac{m}{2}}$, and $\lambda_{j}:=\lambda_{s,\chi,\sigma_{j}}$ to be the continuous linear functional on $\sigma_{j}$ defined by the local Hecke integral (3.1). Let $\tau_{j}$ be the minimal $K$-type of $\sigma_{j}$. As described in Subsection 3.1, $\tau_{j}$ is a two dimensional space, in which there exists a basis $\{v_{j},v_{-j}\}$ with property that

1. (1)

$v_{-j}=\sigma_{j}(\begin{pmatrix}-1&\\ &1\end{pmatrix})v_{j}=\tau_{j}(\begin{pmatrix}-1&\\ &1\end{pmatrix})v_{j}$; 2. (2)

$\sigma_{j}(\theta)v_{j}=\tau_{j}(\theta)v_{j}=e^{i\cdot(l_{j}+1)\theta}v_{j}$ and $\sigma_{j}(\theta)v_{-j}=\tau_{j}(\theta)v_{-j}=e^{-i\cdot(l_{j}+1)\theta}v_{-j}$; 3. (3)

$\langle\lambda_{j},v_{j}+\chi(-1)v_{-j}\rangle=L(s,\sigma_{j}\otimes\chi)$.

Denote by $\tau_{j}^{\epsilon,\chi}$ the one dimensional subspace of $\tau_{j}$ spanned by the vector $v_{j}+\chi(-1)v_{-j}$. By [26, Proposition 8.1], the minimal $K$-type $\tau$ of $\pi$ is also the minimal $K$-type in the induced representation

[TABLE]

Every function $f\in V_{\pi_{0}}$ is a smooth function in $C^{\infty}(K,\tau_{1}\otimes\cdots\otimes\tau_{n})$ satisfying the equivariant property:

[TABLE]

It may not be convenient to work with vector-valued functions. By using the basis of $\tau_{1}\otimes\tau_{2}\otimes\cdots\otimes\tau_{n}$, we may obtain scalar valued functions as follows: For every $f\in C^{\infty}(K,\tau_{1}\otimes\cdots\otimes\tau_{n})$, we can write

[TABLE]

where the summation is taken over all possible choices $\eta_{j}\in\{\pm j\}$ and $f_{\eta_{1},\eta_{2},\cdots,\eta_{n}}(k)\in C^{\infty}(K)$.

Lemma 4.1**.**

A smooth function $f\in C^{\infty}(K,\tau_{1}\otimes\cdots\otimes\tau_{n})$ with basis expansion (4.5) satisfies the equivariant property (4.4) if and only if both

[TABLE]

and

[TABLE]

hold for all possible choices of $\epsilon_{i}\in\{\pm 1\}$ and $\eta_{j}\in\{\pm j\}$. Consequently, the map $f\mapsto f_{1,2,\cdots,n}$ defines a $K$-module isomorphism between $\pi_{0}$ and ${\mathrm{Ind}}_{(M\cap K)^{0}}^{K}\chi_{l_{1}+1}\otimes\chi_{l_{2}+1}\otimes\cdots\otimes\chi_{l_{n}+1}.$

Proof.

Let us assume that $f$ satisfies the equivariant property (4.4). Then $f(k(\theta_{1},\theta_{2},\cdots,\theta_{n})k)$ is equal to

[TABLE]

By comparing the coefficients of each basis vector $v_{\eta_{1}}\otimes v_{\eta_{2}}\otimes\cdots\otimes v_{\eta_{n}}$, we get (4.6). The equivariant property of $f$ with respect to $c(\epsilon_{1},\cdots,\epsilon_{n})$ can be obtained by the following calculation:

[TABLE]

By comparing the coefficients for each basis vector, we get (4.7). On the other hand, since ${\mathrm{O}}_{2}(\mathbb{R})$ is a semidirect product of ${\mathrm{SO}}_{2}(\mathbb{R})$ and $\mathbb{Z}/2\mathbb{Z}$, once a function $f(k)$ satisfies (4.6) and (4.7), one must have that $f(k)$ satisfies the desired equivariant property (4.4). This completes our proof. ∎

Under the $K$-isomorphism in Lemma 4.1, once we construct a complex-valued function $f_{1,2,\cdots,n}$ in the minimal $K$-type $\tau_{0}$ of induced space

[TABLE]

we can use the equivariant properties (4.6), (4.7) and the basis expansion (4.5) to recover a vector-valued function $f$ in the minimal $K$-type $\tau$ of $\pi_{0}$.

4.2. On certain minimal $K$-type functions

As explained in the previous subsection, we only need to construct a scalar-valued function in the minimal $K$-type of the induced representation $\pi_{\textrm{sc}}$ in (4.8). It turns out that our construction fits in a more general framework that may possibly be useful in other cases. Thus we would like to discuss this general framework in this subsection and provide a detailed formula of the desired cohomological test vector in $\pi_{0}$ (see (4.3)) in the next subsection.

In this Subsection, we set $K^{\prime}$ to be either a compact special unitary group ${\mathrm{SU}}_{m}$ or a compact unitary group ${\mathrm{U}}_{m}$, a compact special orthogonal group ${\mathrm{SO}}_{m}({\mathbb{R}})$ or a compact orthogonal group ${\mathrm{O}}_{m}({\mathbb{R}})$, or a compact symplectic group ${\mathrm{Sp}}(2m)$, for any integer $m\geq 1$. Or even more generally we may take $K^{\prime}$ to be a finite product of those compact Lie groups. We fix once and for all a maximal torus $T^{\prime}$ of $(K^{\prime})^{0}$, and obtain a positive root system $\Phi^{+}((\mathfrak{k}^{\prime})^{\mathbb{C}},(\mathfrak{t}^{\prime})^{\mathbb{C}})$. It is well-known that all irreducible representations of $(K^{\prime})^{0}$ can be parameterized by the highest weights, by the standard highest weight theory of compact groups [15, Theorem 5.110]. Such a highest weight can be identified with a certain number of integers in the decreasing order.

Remark 4.2**.**

To be precise, the highest weights which we used in this paper are analytically integral.

When passing from $(K^{\prime})^{0}$ to $K^{\prime}$, we need to clarify our parametrization of irreducible representations of each disconnected factor of $K^{\prime}$. It is enough to clarify the ’highest weight’ of a irreducible representation of ${\mathrm{O}}_{m}({\mathbb{R}})$ $(m=1,2,\cdots)$, as they are the only disconnected simple compact Lie groups among the list of compact groups we considered in the previous paragraph.

For odd orthogonal groups ${\mathrm{O}}_{2l+1}({\mathbb{R}})$ ($l=0,1,2\cdots$), as

[TABLE]

any irreducible ${\mathrm{O}}_{2l+1}({\mathbb{R}})$-module $\sigma$ is parameterized by $\mu[\xi]$, where $\mu=(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{l})\in{\mathbb{Z}}^{l}$ is the highest weight of $\sigma_{0}$ when restricted on ${\mathrm{SO}}_{2l+1}({\mathbb{R}})$; and $\xi={\mathrm{id}}$ or ${\mathrm{sgn}}$ is a quadratic character of the component group ${\mathrm{O}}_{2l+1}({\mathbb{R}})/{\mathrm{SO}}_{2l+1}({\mathbb{R}})$. We call $\nu=\mu[{\mathrm{id}}]$ or $\mu[{\mathrm{sgn}}]$ to be the highest weight of $\sigma$.

Remark 4.3**.**

In particular, when $l=0$, we just say that $[{\mathrm{id}}]$ or $[{\mathrm{sgn}}]$ are highest weights of the one-dimensional representation of the group ${\mathrm{O}}_{1}({\mathbb{R}})={\mathbb{Z}}/2{\mathbb{Z}}.$

For even orthogonal groups ${\mathrm{O}}_{2l}({\mathbb{R}})$ ($l=1,2\cdots$), the restriction of an irreducible ${\mathrm{O}}_{2l}({\mathbb{R}})$-module $\sigma$ to ${\mathrm{SO}}_{2l}({\mathbb{R}})$ is either irreducible, or reducible with two irreducible summands. If $\sigma_{0}:=\sigma\lvert_{{\mathrm{SO}}_{2l}({\mathbb{R}})}$ is irreducible, then the highest weight of $\sigma_{0}$ has the form

[TABLE]

In this case, there are exactly two non-equivalent irreducible ${\mathrm{O}}_{2l}({\mathbb{R}})$-modules whose restriction to ${\mathrm{SO}}_{2l}({\mathbb{R}})$ has the above highest weight. To distinguish them, we call $\nu:=\mu[\epsilon]$ with $\epsilon=1$ or $-1$ to be the highest weight $\sigma$. If $\sigma_{0}$ is reducible, then it decomposes into two irreducible ${\mathrm{SO}}_{2l}({\mathbb{R}})$-modules with highest weights

[TABLE]

where

[TABLE]

In this case, we call

[TABLE]

to be the highest weight of $\sigma$.

Remark 4.4**.**

We would like to point out that the above summary for orthogonal groups is well-known, for example, see [10, Section 5.5.5].

Accordingly, we have the notion of highest weight vector, and similarly the notion of lowest weight vector of an irreducible representation of orthogonal groups. Thus, the notion of highest and lowest weight vectors of $K^{\prime}$ are now clear. When $\beta$ is an irreducible $K^{\prime}$-module with highest weight $\nu$, we define $\chi_{\nu}$ to be the character of $T^{\prime}$ on the one-dimensional space generated by any nonzero highest weight vector of $\beta$.

Next, we will clarify the notion of Cartan component of the tensor product of two irreducible $K^{\prime}$-modules. It suffices to clarify that for the orthogonal groups ${\mathrm{O}}_{m}({\mathbb{R}})$, as the Cartan component of the tensor product of two irreducible representations of connected compact groups is well-known. Suppose that $(\alpha,W_{\alpha})$ and $(\beta,W_{\beta})$ are two irreducible ${\mathrm{O}}_{m}({\mathbb{R}})$-modules with two highest weight vectors $v_{\alpha}\in W_{\alpha},v_{\beta}\in W_{\beta}$. Then there exists a unique irreducible ${\mathrm{O}}_{m}({\mathbb{R}})$-submodule of the tensor product $W_{\alpha}\otimes W_{\beta}$, called the Cartan component of $W_{\alpha}\otimes W_{\beta}$, generated by $v_{\alpha}\otimes v_{\beta}$. It is clear now that one can extend the notion of Cartan component of the tensor product of two irreducible $K^{\prime}$-modules in a natural way.

For any irreducible $K^{\prime}$-module $(\beta,W)$ with highest weight $\nu$, we write $(\beta^{*},W^{*})$ for its dual representation and $\nu^{*}$ for the highest weight of $\beta^{*}$. It is well-known that as a $K^{\prime}$-module on the left, the space of $K^{\prime}$-finite vectors $C^{\infty}(K^{\prime})_{\textrm{fin}}$ is completely reducible, and

[TABLE]

where $\widehat{K^{\prime}}$ stands for the unitary dual of $K^{\prime}$ as usual. Under the $K^{\prime}$-action of $C^{\infty}(K^{\prime})_{{\mathrm{fin}}}$ on the right, the space ${\mathrm{Hom}}_{K^{\prime}}(\alpha,C^{\infty}(K^{\prime})_{\textrm{fin}})$ carries a natural $K^{\prime}$-action such that

[TABLE]

as $K^{\prime}$-modules. We summarize the above well-known result of representation of compact groups as the following Lemma, which will be used repeatedly in this Section.

Lemma 4.5**.**

Suppose that under the left action of $K^{\prime}$, a function $f(x)\in C^{\infty}(K^{\prime})$ generates an irreducible $K^{\prime}$-submodule $\alpha$ of $C^{\infty}(K^{\prime})$, then under the right $K^{\prime}$-action, it generates an irreducible $K^{\prime}$-submodule of $C^{\infty}(K^{\prime})$ which is isomorphic to the dual representation $\alpha^{*}$.

Suppose further that we have a closed subgroup $C^{\prime}$ of $K^{\prime}$.

Proposition 4.6**.**

Let $\lambda_{W,\textrm{endo}}$ be a $C^{\prime}$-invariant linear functional on $W^{*}\otimes W$, i.e.

[TABLE]

for any $c^{\prime}\in C^{\prime}$, $v\in W^{*}\otimes W.$ Then for any highest weight vectors $w\in W$, $w^{*}\in W^{*}$, the following function

[TABLE]

is right $C^{\prime}$-invariant and lives in the induced representation ${\mathrm{Ind}}_{T^{\prime}}^{K^{\prime}}\chi_{\nu}^{-1}\chi_{\nu^{*}}^{-1}$. Moreover, if $f_{W}$ is nonzero, it generates a minimal $K^{\prime}$-type $\tau^{\prime}$ of the induced representation ${\mathrm{Ind}}_{T^{\prime}}^{K^{\prime}}\chi_{\nu}^{-1}\chi_{\nu^{*}}^{-1}$, in which $f_{W}$ is a highest weight vector. In this case, the minimal $K^{\prime}$-type $\tau^{\prime}$ is isomorphic to the Cartan component of $W^{*}\otimes W$.

Proof.

Right $C^{\prime}$-invariance of $f_{W}$ follows directly from the $C^{\prime}$-invariance of the linear functional $\lambda_{W,\,\textrm{endo}}$. For any $t\in T^{\prime}$, we have

[TABLE]

Thus $f_{W}(x)\in{\mathrm{Ind}}_{T^{\prime}}^{K^{\prime}}\chi_{\nu}^{-1}\chi_{\nu^{*}}^{-1}.$ By the Frobenius reciprocity law, the Cartan component of $W^{*}\otimes W$ is isomorphic to a minimal $K$-type of ${\mathrm{Ind}}_{T^{\prime}}^{K^{\prime}}\chi_{\nu}^{-1}\chi_{\nu^{*}}^{-1}$. It remains to check the last claim in the Proposition.

Let $w_{1}\in W$ and $w_{1}^{*}\in W^{*}$ be any two vectors. We define a smooth function on $K^{\prime}$ by

[TABLE]

It is straightforward to check that under the left $K^{\prime}$-action of $C^{\infty}(K^{\prime})$, the map

[TABLE]

is a $K^{\prime}$-intertwining operator. Together with (4.11), this implies that if $f_{W}(x)$ defined in (4.10) is nonzero, then under the left action of $K^{\prime}$, it generates an irreducible $K^{\prime}$-submodule of $C^{\infty}(K^{\prime})$ isomorphic to the Cartan component of $W^{*}\otimes W$. Moreover, $f_{W}(x)$ is a highest weight vector in this irreducible $K^{\prime}$-submodule. By Lemma 4.5, under the right $K^{\prime}$-action, $f_{W}(x)$ generates an irreducible $K^{\prime}$-submodule of $C^{\infty}(K^{\prime})$ isomorphic to dual of the Cartan component of $W^{*}\otimes W$. The Proposition finally follows from the simple observation that $W^{*}\otimes W$ is a self-dual $K^{\prime}$-module. ∎

Similarly, we have the following Corollary, whose proof is the same as that of Proposition 4.6, and will be omitted here.

Corollary 4.7**.**

Given $n$ irreducible $K^{\prime}$-module $W_{1},W_{2},$ $\cdots W_{n}$, for each $j=1,2,\cdots,n$, let $\nu_{j}$ be the highest weight of $W_{j}$ and a $C^{\prime}$-invariant function $f_{W_{j}}(x)$ be as in (4.10). For $n$ non-negative integers $m_{1},m_{2},\cdots,m_{n}$, define a character of $T^{\prime}$ via

[TABLE]

Then the smooth function

[TABLE]

is a $C^{\prime}$-invariant function in the induced representation ${\mathrm{Ind}}_{T^{\prime}}^{K^{\prime}}\chi_{\nu}^{-1}$. Moreover, if $f(x;W_{1},W_{2},\cdots,W_{n})$ is nonzero, it generates a minimal $K^{\prime}$-type $\tau^{\prime}$ of ${\mathrm{Ind}}_{T^{\prime}}^{K^{\prime}}\chi_{\nu}^{-1},$ in which $f(x;W_{1},W_{2},\cdots,W_{n})$ is a highest weight vector. In this case, the minimal $K^{\prime}$-type $\tau^{\prime}$ is isomorphic to the Cartan component of

[TABLE]

4.3. On $K\cap H$-equivariant cohomological test vectors

Let us retain the notation in Subsection 4.1. Now we are fully ready to construct explicitly a cohomological test vector in the minimal $K$-type $\tau$ of $\pi$, with desired properties.

Set $K^{\prime}=K={\mathrm{O}}_{2n}({\mathbb{R}})$ and $C^{\prime}=H\cap K={\mathrm{O}}_{n}({\mathbb{R}})\times{\mathrm{O}}_{n}({\mathbb{R}}),$ where $n\geq 2$ is an integer. The maximal torus $T^{\prime}$ in the previous subsection is chosen to be $(M\cap K)^{0}$, which is isomorphic to a product of $n$ copies of ${\mathrm{SO}}_{2}({\mathbb{R}})$. We consider the standard representation of $K^{\prime}$ on $V={\mathbb{C}}^{2n}$, where every vector in $V$ is realized as a column vector and the group $K$ acts by multiplication on the left. Take a standard basis $e_{1},e_{2},\cdots,e_{2n}$ in $V$. Then the dual representation $V^{*}$ can be realized as the space of row vectors with $2n$ entries, where an element $x\in K$ acts via multiplication by $x^{-1}$ on the right. We take $e_{1}^{t},e_{2}^{t},\cdots e_{2n}^{t}$ to be the dual basis of $V^{*}$. For each $j=1,2,\cdots,n$,

[TABLE]

are weight vectors of $V$ and $V^{*}$ respectively with weight

[TABLE]

where $1$ locates in the $j$-th position.

For each $j=1,2,\cdots,n$, we consider the fundamental representation $\beta_{j}$ of $K$ on the space $W_{j}:=\wedge^{j}V$. By [10, Theorem 5.5.13], when $j\neq n$, the restriction of $W_{j}$ on $K^{0}$ is irreducible with highest weight

[TABLE]

where all the $1$’s locate in the first $j$ positions; when $j=n$, $W_{n}$ is only irreducible as a $K$-module (not a $K^{0}$-module) whose highest weight is

[TABLE]

It is clear that the following linear functional on $V^{\ast}\otimes V$ is $H\cap K$-invariant:

[TABLE]

Thus, the $H\cap K$-invariant linear functional $\lambda_{V,\textrm{endo}}$ induces an $H\cap K$-invariant linear functional $\lambda_{W_{j},\textrm{endo}}$ on $W_{j}^{*}\otimes W_{j}$ defined as follows: for every $v_{1},v_{2},\cdots,v_{j}\in V$, $v_{1}^{*},v_{2}^{*},\cdots,v_{j}^{*}\in V^{*}$, we define

[TABLE]

where $S_{j}$ is the symmetric group which permutes $j$ symbols, and ${\mathrm{sgn}}(s)$ is the sign of the permutation $s$, i.e. it is $1$ if $s$ is an even permutation, and it is $-1$ if $s$ is an odd permutation.

We take a highest weight vector of $W_{j}$:

[TABLE]

and a highest weight vector of $W_{j}^{*}$:

[TABLE]

where all $u_{1},u_{2},\cdots,u_{j},u_{1}^{*},u_{2}^{*},\cdots,u_{j}^{*}$ are defined in (4.13).

Corollary 4.8**.**

For each $j=1,2,\cdots,n$, the following smooth function on $K$ defined by

[TABLE]

is right $H\cap K$-invariant, and lives in the induced representation

[TABLE]

Moreover,

[TABLE]

where $w$ is the Weyl element defined in (3.4). Under the left $K$-action, $f_{W_{j}}(x)$ is a highest weight vector; and under the right $K$-action, it generates a minimal $K$-type of $W_{{\mathrm{even}},j}.$

Proof.

Only the equation $f_{W_{j}}(w)=1$ needs to be checked, the other statements follow directly from Proposition 4.6. Let $w$ be the Weyl element in (3.4). Then

[TABLE]

By a direct matrix computation, it is easy to check that for any $1\leq l\leq j$,

[TABLE]

where $\delta_{l}^{k}$ is the standard Kronecker delta symbol. Thus, $f_{W_{j}}(w)=1$. ∎

Corollary 4.9**.**

Given a sequence of decreasing even integers

[TABLE]

for each $j=1,2,\cdots,n$, let smooth functions $f_{W_{j}}(x)$ be constructed as in (4.17). Then the function

[TABLE]

is right $K\cap H$-invariant, and lives in the induced representation

[TABLE]

Moreover, we have

[TABLE]

where $w$ is the Weyl element defined in (3.4). Under the left $K$-action, $F_{wt,\vec{N^{\prime}}}(x)$ is a highest weight vector; and under the right $K$-action, it generates a minimal $K$-type of $W_{{\mathrm{even}},\vec{N^{\prime}}}.$

Proof.

The fact that $F_{wt,\vec{N^{\prime}}}(w)=1$ follows directly from Corollary 4.8. Thus, Corollary 4.9 follows directly from Corollary 4.7. ∎

Recall that $\chi$ and $\omega$ are characters of ${\mathbb{R}}^{\times}$, $\chi_{0}$ and $\omega_{0}$ are the restrictions of the characters $\chi$ and $\omega$ on $\{\pm 1\}$. Our goal is to construct a ${\mathbb{C}}$-valued function in the minimal $K$-type $\tau_{0}$ of $\pi_{\textrm{sc}}$ (defined in (4.8)) satisfying the right $K\cap H$-equivariance as in (3.11). In the special case $\omega_{0}=\chi_{0}={\mathrm{id}}$, we set $N_{j}^{\prime}=l_{j}+1$ ($j=1,2,\cdots,n$). Then $\vec{N^{\prime}}=(N_{1}^{\prime},N_{2}^{\prime},\cdots,N_{n}^{\prime})$ is a decreasing sequence of positive integers. By Corollary 4.9, the function $F_{wt,\vec{N^{\prime}}}(x)$ constructed therein is exactly the right $K\cap H$-invariant vector in the minimal $K$-type $\tau_{0}$ of $\pi_{\textrm{sc}}$. To take care of the other cases, i.e. one of $\chi_{0},\omega_{0}$ is non-trivial, we introduce two more functions.

Proposition 4.10**.**

Let $u_{j}^{*}$ be as in (4.13) and the Weyl element $w$ be defined in (3.4). Define the following two functions on $K$:

[TABLE]

and

[TABLE]

Then $F_{\mathrm{left}}(w)=F_{\mathrm{right}}(w)=1$. Under the left $K$-action, both $F_{\mathrm{left}}(x)$ and $F_{\mathrm{right}}(x)$ are highest weight vectors, while under the right $K$-action, they both generate the minimal $K$-type of the induced representation

[TABLE]

Proof.

By a straightforward matrix computation, one can check that $F_{\mathrm{left}}(w)=F_{\mathrm{right}}(w)=1$ and that both $F_{\mathrm{left}}(x)$ and $F_{\mathrm{right}}(x)$ live in the induced representation $W_{\textrm{odd}}$. It remains to show that $F_{\mathrm{left}}(x)$ generates the minimal $K$-type of $W_{\textrm{odd}}$. The proof for $F_{\mathrm{right}}(x)$ is exactly the same.

Note that $u_{1}^{*}\wedge u_{2}^{*}\wedge\cdots\wedge u_{n}^{*}$ is a highest weight vector in the fundamental representation $\bigwedge^{n}V^{*}$. Under the left $K$-action, $F_{\mathrm{left}}(x)$ is a highest vector and generate an irreducible submodule of $C^{\infty}(K^{\prime})$ isomorphic to $\bigwedge^{n}V^{*}$. By Lemma 4.5, under the right $K$-action, $F_{\mathrm{left}}(x)$ generates an irreducible $K$-submodule of $C^{\infty}(K^{\prime})$ isomorphic to $\bigwedge^{n}V$. Since the representation $\bigwedge^{n}V$ has highest weight $[1,1,\cdots,1,1]$, it is isomorphic to the minimal $K$-type of $W_{\textrm{odd}}$. We are done. ∎

Definition 4.11**.**

Let $\vec{N}:=(N_{1},N_{2},\cdots,N_{n})$ be a sequence of positive integers with the same parity in the decreasing order and $\omega$ be a character of ${\mathbb{Z}}/2{\mathbb{Z}}$. We say $\vec{N}$ is parity-compatible with $\omega_{0}$ if when $N_{1}$ is even, $\omega_{0}$ is trivial; and when $N_{1}$ is odd, $\omega_{0}$ is the sign character.

For any decreasing sequence $\vec{N^{\prime}}:=(N_{1}^{\prime},N_{2}^{\prime},\cdots,N_{n}^{\prime})$ of non-negative even integers, recall the function $F_{wt,\vec{N^{\prime}}}(x)$ as constructed in Corollary 4.9 and the functions $F_{\mathrm{left}}(x)$, $F_{\mathrm{right}}(x)$ as constructed in Proposition 4.10. Now we are ready to construct the right $(K\cap H)$-equivariant cohomological test vector in the induced representation $\pi_{\textrm{sc}}$ defined in (4.8), based on these three types of functions.

Theorem 4.12**.**

Let $\vec{N}:=(N_{1},N_{2},\cdots,N_{n})$ be a sequence of positive integers with the same parity in the decreasing order, and $\omega_{0}$ be a character of ${\mathbb{Z}}/2{\mathbb{Z}}$ that is parity-compatible with $\vec{N}$, as in Definition 4.11. For any character $\chi_{0}$ of ${\mathbb{Z}}/2{\mathbb{Z}}$, define a function $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)\in C^{\infty}(K)$ as follows:

1. (1)

If $N_{1}$ is even and $\chi_{0}$ is trivial, set $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x):=F_{wt,\vec{N^{\prime}}}(x),$ where $N_{j}^{\prime}=N_{j}$, $j=1,2,\cdots,n$. 2. (2)

If $N_{1}$ is even and $\chi_{0}$ is the sign character, set

[TABLE]

where $N_{j}^{\prime}=N_{j}-2$, $j=1,2,\cdots,n$. 3. (3)

If $N_{1}$ is odd and $\chi_{0}$ is trivial, set

[TABLE]

where $N_{j}^{\prime}=N_{j}-1$, $j=1,2,\cdots,n$. 4. (4)

If $N_{1}$ is odd and $\chi_{0}$ is the sign character, set

[TABLE]

where $N_{j}^{\prime}=N_{j}-1$, $j=1,2,\cdots,n$.

Then $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ lives in the minimal $K$-type of

[TABLE]

and satisfies the right $K\cap H$-equivariant property (3.11). Moreover, $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(w)=1$, where $w$ is the Weyl element defined in (3.4).

Proof.

The fact that $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ lives in

[TABLE]

and $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(w)=1$ follows directly from Corollary 4.9 and Proposition 4.10. The right $(K\cap H)$-equivariance can be checked in an ad-hoc way by direct matrix computations, since $F_{wt,\vec{N^{\prime}}}(x)$ is right $K\cap H$-invariant. Here we only prove that $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ generates the minimal $K$-type of

[TABLE]

Note that $F_{wt,\vec{N^{\prime}}}(x)$ constructed in Corollary 4.9 and the functions $F_{\mathrm{left}}(x)$, $F_{\mathrm{right}}(x)$ constructed in Proposition 4.10 are all highest weight vectors under the left $K$-action. Under the left $K$-action, the function $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ is also a highest weight vector and generates an irreducible $K$-submodule of $C^{\infty}(K)$ with highest weight $(N_{1},N_{2},\cdots,N_{n}).$ The irreducible $K$-module with highest weight $(N_{1},N_{2},\cdots,N_{n})$ is self-dual. By Lemma 4.5, under the right $K$-action, $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ generates an irreducible $K$-submodule with highest weight $(N_{1},N_{2},\cdots,N_{n})$, which is exactly the minimal $K$-type of the induced representation

[TABLE]

∎

It is clear that once we set $N_{j}:=l_{j}+1$, the function $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(x)$ constructed in the above theorem must live in the minimal $K$-type of the induced representation $\pi_{\textrm{sc}}$ as defined in (4.8). Hence, combining with the reduction steps in Subsection 4.1, we obtain the following direct corollary of Theorem 4.12.

Corollary 4.13**.**

Let $\chi_{0},\omega_{0}$ be the restriction of $\chi,\omega$ on $\{\pm 1\}$ as in (3.11). Let $N_{j}:=l_{j}+1$ and $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}$ be the function constructed in Theorem 4.12. Define $f_{1,2,\cdots,n}=\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}$ and $f_{\eta_{1},\eta_{2},\cdots,\eta_{n}}(k)$ by the rules (4.7). Then, as defined in (4.5), the function

[TABLE]

with the summation being taken over all possible choices $\eta_{j}\in\{\pm j\}$, belongs to the minimal $K$-type $\tau$ of $\pi$. Moreover, $f(k)$ satisfies the desired equivariant property (3.11).

Remark 4.14**.**

By the above construction, $f(k)$ depends on $\chi$ and $\omega$.

5. Non-vanishing of Archimedean Local Integrals

With the explicit construction of the cohomological vector (see Subsection 4.3), we can analyze the local integrals $Z(v,s,\chi)$ (defined in (1.16)) and $\Lambda_{s,\chi}(v)$ (defined in (3.10)) in detail. We retain all notation in Subsection 4.3.

Theorem 5.1**.**

Let $f(k)$ be the cohomological vector constructed in Corollary 4.13 and $w$ be the Weyl element defined in (3.4). Then

[TABLE]

As a consequence, as a meromorphic function of $s\in{\mathbb{C}}$, $\frac{1}{L(s,\pi\otimes\chi)}\Lambda_{s,\chi}(f)=1.$ Hence $\Lambda_{s,\chi}$ defines a non-zero element in

[TABLE]

whenever $s$ is not a pole of $L(s,\pi\otimes\chi).$

Proof.

By Theorem 4.12, we have that $\varphi_{\vec{N},\chi_{0}\otimes\chi_{0}\omega_{0}}(w)=1$, and hence

[TABLE]

Recall the matrix $c(\epsilon_{1},\cdots\epsilon_{n})$ from (4.2). By (4.7),

[TABLE]

It follows that by (4.5)

[TABLE]

By Corollary 4.13, $f(k)$ satisfies the desired equivariant property (3.11). As a direct consequence of (3.12) and Corollary 3.2, the following holds for $s\in{\mathbb{C}}$ with sufficiently large real parts:

[TABLE]

Here the local $L$-function is obtained from the local Langlands correspondence for ${\mathrm{GL}}_{2n}({\mathbb{R}})$, which we refer to [17] (also see [16] and [11]). Thus, by meromorphic continuation, for all $s\in{\mathbb{C}},$ $\frac{1}{L(s,\pi\otimes\chi)}\Lambda_{s,\chi}(f)=1.$ We are done. ∎

It is finally clear that Theorem 1.2 holds. As a consequence, we can also show the non-vanishing of the archimedean Friedberg-Jacquet integral $Z(v,s,\chi)$.

Corollary 5.2**.**

There exists a holomorphic function $G(s,\chi)$ such that

[TABLE]

As a consequence, if $v=f(k)$ is the cohomological vector constructed in Corollary 4.13, then whenever $s$ is not a pole of $L(s,\pi\otimes\chi)$, $Z(v,s,\chi)$ does not vanish.

Proof.

By [5, Theorem B], for all but countably many $s\in{\mathbb{C}}$ where $L(s,\pi\otimes\chi)$ does not have a pole, one has

[TABLE]

Since for such pair $(s,\chi)$, both $Z(v,s,\chi)$ and $\Lambda_{s,\chi}$ defines a non-zero element in

[TABLE]

there exists a constant $C(s,\chi)$ (depending on $s$ and $\chi$) such that

[TABLE]

Since both $Z(v,s,\chi)$ and $\Lambda_{s,\chi}$ are meromorphic in $s$ and $\chi$, $C(s,\chi)$ is also meromorphic in $s,\chi$. Now we plug $v=f(k)$ (the cohomological vector constructed in Corollary 4.13) in (5.1), and by Theorem 5.1, we obtain

[TABLE]

Thus, by [4, Theorem 3.1], $C(s,\chi)$ must be holomorphic. Similarly, also by [4, Theorem 3.1], we can choose a smooth vector $v_{0}$ such that $Z(v_{0},s,\chi)=L(s,\pi\otimes\chi)$. Thus, using the same argument as above,

[TABLE]

must also be holomorphic, by Corollary 3.3. Hence $C(s,\chi)$ have no zeroes. This implies that there exists a holomorphic function $G(s,\chi)$ such that $C(s,\chi)=e^{G(s,\chi)}$. ∎