L-functions and theta correspondence for classical groups

Abstract

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.

Appendices

Appendix A: The archimedean theory

In this appendix we will let F be an archimedean local field except in the statement of Lemma A.2. Since the sections constructed in the proofs of Lemma 6.2 and Theorem 7.1 may not be $\bar{K}^{\Box}$-finite, a complication arises in the archimedean case. To get around this problem, we redefine the L-factor by using general smooth sections rather than $\bar{K}^{\Box}$-finite sections. To show that the two definitions agree, we shall show that the ratio of the local integral divided by this L-factor is continuous in the appropriate topology. Kudla and Rallis [26] have carried this out for symplectic and orthogonal groups. The subtle convergence arguments can be simplified considerably by applying the inductive argument developed in Sect. 5 to smooth sections.

A character χ of E ^× determines a holomorphic family of characters χ _ψ,s ∘Δ of $\bar{P}$ and I ^∞(s,χ) stands for the smooth degenerate principal series representation. Since P∖G ^□=M∩K ^□∖K ^□, all representations I ^∞(s,χ) can be realized on the same space I ^∞(χ) of smooth functions f on $\bar{K}^{\Box}$ which satisfy f(pg)=χ′(Δ(p))f(g) for $g\in\bar{K}^{\Box}$ and $p\in\bar{M}\cap\bar{K}^{\Box}$. Here χ′=χ _ψ in Case ($\mathrm{I}_{4}^{m}$) and χ′=χ in the other cases. Note that I ^∞(χ) is a Fréchet space with respect to the seminorms ∥Xf∥, where $\|f\|=\max_{k\in\bar{K}^{\Box}}|f(k)|$ and X ranges over the universal enveloping algebra $U(\mathfrak{k}^{\Box})$ of the complexified Lie algebra $\mathfrak{k}^{\Box}$ of K ^□. In this way we can view I ^∞(s,χ) as a holomorphic family of representations.

A holomorphic section f ^(s) of I ^∞(s,χ) is a holomorphic function on $\mathbb{C}$ with values in I ^∞(χ). For each $g\in\bar{G}^{\Box}$ we define f ^(s)(g) by writing g=pk with $p\in\bar{P}$ and $k\in\bar{K}^{\Box}$, and taking

$$f^{(s)}(g)=\chi_{\psi,s+\delta\rho_n/2} \bigl(\Delta (p) \bigr)f^{(s)}(k). $$

As in the $\bar{K}^{\Box}$-finite setting, we have the concept of good sections in the smooth setting. The obvious adaptation of the results stated in Sects. 3.5 and 5.2 still holds for good sections of I ^∞(s,χ). However, it should be remarked that the proof of Lemma 6.2 is more involved in the archimedean case. Theorem 4.1 of [3] states that principal series representations admit a Bruhat filtration, and it is possible to extend this work to I ^∞(s,χ), which allows us to argue in the archimedean case in the same way as in the p-adic case.

Lemma A.1

For any $\pi\in\mathrm{Irr}(\bar{G})$ and any character χ of E ^× there exists a local Euler factor L(s) such that for any ξ∈π ^∨⊠π and any good section f ^(s) of I ^∞(s,χ) the quotient Z(ξ,f ^(s))/L(s+1/2) is entire.

Proof

We can use Propositions 4.2 and 5.3 to reduce the statement to the case where φ is anisotropic or the case where n=1 and φ=0. The former case is evident. In the latter case G=GL₁(C), G ^□=GL₂(C) and the zeta integral is easy to compute (cf. [32]). The embedding of G×G into G ^□ is given by , where . Then P is the group of matrices whose lower left n×n blocks are zero. Put C ¹={z∈C ^× | |z|=1}. Then $C=\mathbb{R}^{\times}_{+}\cdot C^{1}$. By twisting π by an unramified character we may assume that χ _π is trivial on $\mathbb{R}^{\times}_{+}$. Let α _i be complex numbers such that $\chi_{i}(a)=a^{\alpha _{i}}$ for $a\in\mathbb{R}^{\times}_{+}$ and i=1,2. The zeta integral is

$$\begin{aligned} &\int_{C^\times}H_\xi(g)f^{(s)} \biggl(w_1 \begin{pmatrix} {g} & {0} \\ {0} & {1} \end{pmatrix} \biggr)dg \\ &\quad =\int_{C^1}H_\xi(c) \int _{\mathbb{R}^\times_+}f^{(s)} \biggl(w_1 \begin{pmatrix} {ac} & {0} \\ {0} & {1} \end{pmatrix} \biggr) d^\times adc. \end{aligned}$$

The Iwasawa decomposition for in $\mathrm{GL}_{2}(\mathbb{R})$ is

$$w_1 \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a(a^2+1)^{-1/2} & * \\ 0 & (a^2+1)^{1/2} \end{pmatrix} \kappa _\theta, $$

where

$$\begin{aligned} \kappa _\theta&= \begin{pmatrix} \cos\theta& -\sin\theta\\ \sin\theta& \cos \theta \end{pmatrix} , \quad \tan\theta=a \quad (0<\theta<\pi/2). \end{aligned}$$

If we write the translate of f ^(s) by for $f_{c}^{(s)}$, then the inner integral is

$$\begin{aligned} &\int_{\mathbb{R}^\times_+} \biggl(\frac{a}{a^2+1} \biggr)^{s+\delta /2}a^{\alpha _1} \bigl(a^2+1 \bigr)^{-(\alpha _1+\alpha _2)/2}f_c^{(s)}( \kappa _\theta)d^\times a \\ &\quad =\int^{\pi/2}_0(\sin\theta)^{s+\alpha _1-1+\delta/2}(\cos \theta)^{s+\alpha _2-1+\delta/2}f_c^{(s)}(\kappa _\theta)d \theta. \end{aligned}$$

Put s _i =s+α _i −1+δ/2. This integral converges, provided that ℜs>−min{ℜα ₁,ℜα ₂}−δ/2. We split the domain of integration into two parts. Integration by parts gives us

$$\begin{aligned} &\int^{\pi/4}_0(\sin\theta)^{s_1}(\cos \theta)^{s_2}f_c^{(s)}(\kappa _\theta)d \theta \\ &\quad =\frac{1}{s_1+1} \biggl(2^{-(s_1+s_2)/2}f^{(s)}_c( \kappa _{\pi/4})-\int^{\pi/4}_0(\sin \theta)^{s_1+1}h(s,\theta,c)d\theta \biggr), \end{aligned}$$

where $h(s,\theta,c)=\frac{\partial}{\partial\theta}\{(\cos \theta )^{s_{2}-1}f_{c}^{(s)}(\kappa _{\theta})\}$. Likewise for the integral over π/4≤θ≤π/2. This allows us to extends Z(ξ,f ^(s)) meromorphically to the region ℜs>−min{ℜα ₁,ℜα ₂}−δ/2−1. We can again integrate by parts and eventually we see that it continues to a meromorphic function in s on the whole plane. Moreover, the poles of all the local integrals are contained in the poles of Γ(s+α ₁+δ/2)Γ(s+α ₂+δ/2) with multiplicity. □

For $\phi\in C^{\infty}_{c}(\bar{G})$ we can define a section $f^{(s)}_{\phi}$ of I ^∞(s,χ) by requiring that $\mathrm{supp}(f^{(s)}_{\phi})\subset \bar{P}\cdot(\bar{G}\times\bar{G})$ and $f^{(s)}_{\phi}((g,e))=\phi(g)$ for $g\in\bar{G}$. The following lemma can be proven by choosing ϕ to be supported in a small neighborhood.

Lemma A.2

([26, 49])

For given $\pi\in\mathrm{Irr}(\bar{G})$ there is a choice of ξ∈π ^∨⊠π and $\phi\in C^{\infty}_{c}(\bar{G})$ such that $Z(\xi ,f_{\phi}^{(s)})=1$.

Now we choose a meromorphic function L(s) in such a way that it not only cancels all poles of the local integrals for good sections of I ^∞(s,χ), but also dividing by it introduces no extraneous zeros. From Lemma A.2 L(s) vanishes nowhere. The following lemma concludes that L(s) is determined by the poles of the family of local integrals even using good sections of I(s,χ).

Lemma A.3

L(s,π ^∨×χ)/L(s) is an invertible function.

Proof

Fix $s^{\prime}\in\mathbb{C}$. Our task is to show that Z(ξ,f ^(s))/L(s+1/2) does not have a zero at s=s′ for a suitable choice of ξ∈π ^∨⊠π and a good section f ^(s) of I(s,χ). For each h∈I ^∞(χ) and $s\in\mathbb{C}$ we define a function $h_{s}:\bar{G}^{\Box}\to\mathbb{C}$ by

$$\begin{aligned} h_s(pk)&=\chi_{\psi,s+\delta\rho_n/2} \bigl(\Delta (p) \bigr)h(k) \quad \bigl(p \in \bar{P},\; k\in\bar{K}^\Box \bigr). \end{aligned}$$

By the argument of [65], for any $X\in U(\mathfrak{g}^{\Box})$ and for s in a compact set, there is a seminorm μ such that ∥Xh _s ∥≤μ(h) for all h∈I ^∞(χ).

As we observed in the course of the proof of Lemma A.1, if n=1 and φ=0, then for ξ∈π ^∨⊠π and for s in a compact set, there exist a polynomial Q(s) and a seminorm μ such that for all h∈I ^∞(χ)

$$ \bigl|Q(s)Z(\xi,h_s)\bigr|\leq\mu(h). $$

(A.1)

Lapid and Rallis [32] proved the functional equation of Theorem 4.1(3) for smooth but not necessarily $\bar{K}^{\Box}$-finite sections. The functional equation reads as in Theorem 5.2 but with $L^{\mathcal{U}}_{\psi}(s,\pi^{\vee}\times\chi)$ replaced by L(s). Since the constant of proportionality is an exponential factor in view of the definition of L(s), we may suppose that $\Re s^{\prime }>-\frac{1}{2}$. In this region the intertwining operator Ψ(s,χ) appearing in Proposition 4.2 is absolutely convergent by Lemma 5.1, and [65, Lemma 10.1.11] states that Ψ(s,χ) is a bounded operator with respect to ∥⋅∥, independently of s. We thus obtain an estimate similar to (A.1) in general. Let l be the order of the pole of L(s+1/2) at s=s′. Using the Cauchy’s integral formula, we can find a seminorm μ such that for all h∈I ^∞(χ)

$$\Bigl\vert\lim_{s\to s^{\prime}}\bigl(s-s^{\prime}\bigr)^lZ( \xi,h_s) \Bigr\vert\leq\mu(h). $$

Take a holomorphic section h ^(s) of I ^∞(s,χ) so that the limit lim _s→s′(s−s′)^l Z(ξ,h ^(s))≠0. Since this limit depends only on h ^(s′), we may assume that h ^(s) is of the form h _s for some h∈I ^∞(χ). Lemma A.3 now follows from the fact $\bar{K}^{\Box}$-finite vectors are dense in I ^∞(χ). □

We conclude this appendix by indicating how we must modify the arguments in the proofs of Theorems 6.1 and 7.1 in the archimedean case. Our explanation focuses on Theorem 7.1 as the proof of Theorem 6.1 can be modified in an easier manner.

Take $\eta_{1}\in\varrho _{1}^{\vee}\boxtimes\varrho _{1}$ and $\phi _{1}\in C^{\infty}_{c}(N_{-}^{Y})$ in such a way that the quotient $Z(\eta_{1},f^{(s)}_{\phi _{1}})/L^{GJ} (s+\frac{1}{2},\varrho _{1}\otimes\rho(\chi) )$ does not have s=s′. As observed in the proof, one can always find η∈ϱ ^∨⊠ϱ and a holomorphic section f ^(s) of I ^∞(s,χ) which satisfy

$$A \bigl(\eta,f^{(s)} \bigr)=Z \bigl(\eta_1,f^{(s)}_{\phi_1} \bigr). $$

However, f ^(s) may not be $\bar{K}\times\bar{K}$ right finite, and hence we cannot directly apply Lemma 7.6(2). One can check that A(η,h _s ) can be meromorphically continued to the region $\Re s>-\frac{1}{2}$ and that the map

$$A_\eta:h\mapsto\lim_{s\to s^{\prime}}A(\eta,h_s)/L^{GJ} \biggl(s+\frac{1}{2},\varrho _1\otimes\rho(\chi) \biggr) $$

is continuous on I ^∞(χ) by a careful analysis based on the technique of [26]. See the explanation of this point in Sect. 6 of [49]. Choose a $\bar{K}^{\Box}$-finite element h in I ^∞(χ) such that A _η (h)≠0. Choosing ξ∈π ^∨⊠π so that Z(ξ,h _s )=A(η,h _s ), we have completed our proof.

Appendix B: Local factors for classical groups

We recall basic properties of the gamma factors which we need. We refer the reader to [32] for a complete description of the gamma factors and their characterization (cf. [4], where the metaplectic case is discussed).

2.1 B.1 Normalization of the intertwining operator

We begin by recalling the normalization of the intertwining operator. Let F be a local field of characteristic zero. We adopt the notation of Sect. 2. We remind the reader that the algebra D is a direct sum of mutually opposite division algebras in Case (II). We denote by N ⁻ the unipotent radical of the maximal parabolic subgroup of G ^□ stabilizing W ^▽. Let $\mathfrak{g}^{\Box}$, $\mathfrak{n}$, $\mathfrak{n}^{-}$ be the Lie algebras of G ^□, N, N ⁻, respectively. Note that

$$\begin{aligned} \mathfrak{g}^\Box&= \bigl\{X\in\mathrm{End} \bigl(W^\Box,D \bigr)\; |\;\langle xX,y\rangle ^\Box+\langle x,yX\rangle ^\Box=0\text { for }x,y\in W^\Box \bigr\}, \\ \mathfrak{n}&= \bigl\{X\in\mathfrak{g}^\Box\;|\;\mathrm {Ker}X\supset W^\Delta \supset\mathrm{Im}X \bigr\}. \end{aligned}$$

We can identify $\mathfrak{n}$ with the space of ϵ-hermitian forms on W ^□/W ^Δ via (x,y)↦〈xX,y〉^□. We sometimes identify the group N ⁻ with its Lie algebra $\mathfrak{n}^{-}$ via the isomorphism u↦u−I. For $A\in\mathfrak{n}$ we define a unitary character ψ _A of N ⁻ by ψ _A (X)=ψ(τ(XA)). Recall that τ:D→E is the reduced trace. Observe that τ(XA)∈F in all cases. The map A↦ψ _A defines an isomorphism of $\mathfrak{n}$ onto the Pontryagin dual of N ⁻.

For simplicity we exclude the odd orthogonal case and the metaplectic case. The reader can consult [4, 5, 32] for a complete account of modifications required in these cases. We identify W with W ^Δ via x↦(x,x), and W ^□/W ^Δ with W via (x,y)↦x−y. With these identifications the reduced norm ν _W (A)∈E is well-defined. Fix $A\in\mathfrak{n}$. Denote by $\mathbb{C}_{\psi_{A}}$ the one dimensional representation of N ⁻ with action given by ψ _A . For a section f ^(s) of I(s,χ) the integral

$$l_{\psi_A} \bigl(f^{(s)} \bigr)=\int_{N^-}f^{(s)}(u) \psi_A(u)du $$

converges absolutely for ℜs≫0 and defines an N ⁻ equivariant map from I(s,χ) to $\mathbb{C}_{\psi_{A}^{-1}}$. When ν _W (A)∈E ^× and F is a p-adic field, Karel [20] has proven that $l_{\psi_{A}}(f^{(s)})$ admits an entire analytic continuation to the whole s-plane and satisfies a functional equation

$$l_{\psi_A}\circ M(s,\chi)=\chi_s \bigl(\nu_W(A) \bigr)^{-1}c(s,\chi,A,\psi)l_{\psi_A} $$

for some meromorphic function c(s,χ,A,ψ). Analogous results are proven in the archimedean case in [64]. As is proven in [32, Lemma 10], the factor c(s,χ,A,ψ) depends only on the homothety class of A, and for λ∈F ^×

$$ c(s,\chi,A,\psi_\lambda )=\chi_s(\lambda )^{-\delta n}c(s, \chi,A,\psi). $$

(B.1)

Let us put

Lemma B.1

Let C(s,χ,A,ψ) be defined as above. Then it is equal to a(s,χ)/b(−s,ρ(χ)⁻¹) up to multiplication by invertible functions.

Proof

We can find a number field $\mathbb{F}$, a quadratic or trivial extension $\mathbb{E}$ of $\mathbb{F}$, a division algebra $\mathbb{D}$ over $\mathbb {E}$, a place v ₀ of $\mathbb{F}$, a nontrivial additive character ψ′ of $\mathbb {A}_{\mathbb{F}}/\mathbb{F}$, a character χ′ of $C_{\mathbb{F}}$ and an ϵ-hermitian matrix $A^{\prime} \in \mathfrak{n}(\mathbb{F})$ so that $\mathbb{F}_{v_{0}}=F$, $\mathbb {E}_{v_{0}}=E$, $\mathbb{D}_{v_{0}}=D$, $\psi ^{\prime}_{v_{0}}=\psi$, $\chi^{\prime}_{v_{0}}=\chi$ and A′ is equivalent to A over $\mathbb{F}_{v_{0}}$. Let S be a finite set of places of $\mathbb{F}$, containing v ₀, outside of which $\mathbb{E}$, $\mathbb{D}$, ψ′, χ′ and A are all unramified. It follows from (3.1) and [32, (24)] that

$$\frac{a_S(s,\chi^{\prime})}{b_S(s,\chi^{\prime})}\cdot\frac {\beta _S(-s,\rho (\chi^{\prime} )^{-1})}{b_S(-s,\rho(\chi^{\prime})^{-1})}\cdot\prod _{v\in S}c_v\bigl(s, \chi^{\prime}_v,A^{\prime}, \psi^{\prime}_v\bigr)=\frac{\beta _S(s,\chi^{\prime})}{b_S(s,\chi ^{\prime})}, $$

where $\beta _{v}(s,\chi^{\prime}_{v})=L(s+1/2,\chi^{\prime}_{v})$ in Cases (I₃) and ($\mathrm{I}_{4}^{s}$), and $\beta _{v}(s,\chi^{\prime}_{v})=1$ in all other cases. We can rewrite this as

$$ \prod_{v\in S}C_v\bigl(s,\chi ^{\prime}_v,A^{\prime},\psi^{\prime}_v \bigr)=b_S \bigl(-s,\rho\bigl( \chi^{\prime} \bigr)^{-1} \bigr)/a_S\bigl(s,\chi^{\prime}\bigr). $$

(B.2)

This equality translates informally to “$\prod_{v}C_{v}(s,\chi^{\prime} _{v},A^{\prime} ,\psi^{\prime}_{v})=1$”.

We first consider Case (I₄). When F is a p-adic field, the factor c(s,χ,A,ψ) is calculated explicitly by Sweet in [56, 57] (the formula in [56] can be extended to arbitrary characters of F ^×, as was noted in [57]). Applying (B.2) with $\mathbb{F}=\mathbb{Q}$ or $\mathbb {F}=\mathbb{Q}(\sqrt{-1})$ and v ₀=∞, we see that his formula remains true for $F=\mathbb{R}$ or $F=\mathbb{C}$. Lemma B.1 is a simple consequence of this formula.

In the p-adic case an explicit formula for the factor c(s,χ,A,ψ) is obtained in [30] in Case (I₂), and in [67] in Cases (I₀) and (I₁). By the same reasoning, we can check that these formulas are still true in the archimedean case, from which Lemma B.1 follows in Cases (I₀)–(I₂).

Finally, we consider Case (I₃). We can find a number field $\mathbb{F}$ which has two places v ₀ and v ₁ such that $\mathbb{F}_{v_{1}}=\mathbb{F}_{v_{2}}=F$. Let $\mathbb{D}$ be a quaternion algebra with center $\mathbb{F}$ ramified precisely at v ₀ and v ₁. Le us choose ψ′, χ′ and A′ so that $\psi^{\prime} _{v_{i}}=\psi$, $\chi^{\prime}_{v_{i}}=\chi$ and A′ is equivalent to A over $\mathbb{F} _{v_{i}}$ for i=1,2. Proposition 10.4(2) allows us to assume that $\prod_{v\neq v_{1},v_{2}}\eta_{v}(A^{\prime}_{v})=1$. Fix a quadratic form T of dimension 2n over F satisfying χ _T =χ _A . Proposition 10.4(1) gives a global quadratic form which is equivalent to T over $\mathbb{F}_{v_{i}}$ for i=1,2 and whose localizations are $A^{\prime}_{v}$ outside the two places. We deduce from (B.2) that

$$c(s,\chi,A,\psi)^2=c(s,\chi,T,\psi)^2. $$

This finishes the list of remaining cases needed to be considered. □

We are now ready to prove Lemma 3.1.

Proof of Lemma 3.1

The proof mimics the argument of the proof of Theorem on p.106 of [45]. An important ingredient of the proof of [45] is the exact location of the poles of c(s,χ,A,ψ). This is carried out in Lemma B.1, and we can proceed in exactly the same way. □

Definition B.1

The normalization of M(s,χ) is defined by setting

$$M^\dagger_\mathcal{W}(s,\chi,A,\psi)=C(s,\chi,A, \psi)^{-1}M(s, \chi). $$

We can easily see that

$$ M^\dagger_\mathcal{W} \bigl(-s,\rho(\chi)^{-1},A, \psi^{-1} \bigr)\circ M^\dagger_\mathcal{W}(s,\chi,A, \psi)=\mathrm{Id}. $$

(B.3)

2.2 B.2 Local theory via gamma factors

In the ensuing discussion we assume A to be maximally split, i.e., the form (x,y)↦〈xA,y〉^□ on W ^□/W ^Δ admits a totally isotropic subspace of dimension [n/2]. For $\pi\in\mathrm{Irr}(\bar{G})$ we set

$$\gamma^\mathcal{W}(s,\pi\times\chi,\psi)=z(\pi)\varepsilon _{W,\psi}^{-1} \varGamma ^\mathcal{W} \biggl(s- \frac{1}{2},\pi^\vee\times\chi,A, \psi \biggr). $$

Recall that the factor $\varGamma ^{\mathcal{W}}(s,\pi\times\chi ,A,\psi)$ is defined as the proportionality constant of the functional equation in Theorem 4.1(3). See [4, 5, 32] for the definition and description of the gamma factor in the odd orthogonal and metaplectic cases. The gamma factor is independent of the choice of A except in the case G is of type (I₃) and n is odd, in which case it still depends on the choice of maximally split A, but we will suppress the dependence on A from the notation.

The calculation of the gamma factor is done in the archimedean case and minimal cases in [4, 32] except in the quaternion case. These cases are yet to be done. The gamma factor may not be quite correct in these cases, but in this paper we require only that it satisfies the formal properties described in following proposition.

Proposition B.1

(Cf. [4, 32])

The factors $\gamma^{\mathcal{W}}(s,\pi\times\chi,\psi)$ satisfy the following properties.

• (1) If π is a subquotient of $\operatorname{Ind} ^{G}_{P(Y)}\sigma$, then

  $$\gamma^\mathcal{W}(s,\pi\times\chi,\psi)=\gamma^\mathcal {Y}(s, \sigma \times \chi, \psi). $$

• (2) $\gamma^{\mathcal{W}}(s,\pi^{\vee}\times\chi ,\psi )=\gamma^{\mathcal{W}} (s,\pi\times\rho(\chi),\psi)$ except in the metaplectic case, in which case $\gamma^{\mathcal{W}}(s,\pi^{\vee}\times\chi,\psi)=\gamma ^{\mathcal{W}} (s,\pi\times\chi ,\psi^{-1})$.

• (3) $\gamma^{\mathcal{W}}(s,\pi\times\chi,\psi )\gamma^{\mathcal{W}} (1-s,\pi^{\vee}\times\chi^{-1},\psi^{-1})=1$.

• (4) In Case (II) we have

  $$\gamma^\mathcal{W}(s,\pi\times\chi,\psi)=\gamma^{GJ}(s,\pi \otimes \chi_1,\psi)\gamma^{GJ} \bigl(s,\pi^\vee \otimes \chi_2,\psi \bigr). $$

Proof

These results are proven in [4, 32] when δ=1. Propositions 4.1, 4.2 and Lemma 4.2 prove the first assertion. We can prove the second statement in the same way as in [32] (see [4], where the metaplectic case is discussed, and see also the proof of Proposition 5.4). The third statement follows from (B.3). The last statement is proven in the appendix of [70]. □