On completeness of local intertwining periods

Let $F$ be a local field of characteristic zero field with normalized absolute value $|\ |_{F}$. Moreover when $F$ is Archimedean we assume that $F={\mathbb{R}}$, which is not a serious restriction: we see complex reductive groups as real reductive groups by a restriction of scalar argument. Let $G=\mathbf{G}(F)$ be an $F$-reductive group, and let $\theta$ be an involution of $G$ defined over $F$. We denote by $A_{G}$ its split component, i.e. the $F$-points of the connected center of $\mathbf{G}$. For any $A\subseteq G$ we set

$$A^{\theta}=\{a\in A,\ \theta(a)=a\}$$

and

$$A^{\theta,-}=\{a\in A,\ \theta(a)=a^{-1}\}.$$

We set

$$H:=G^{\theta},$$

and call the pair $(G,H)$ reductive symmetric pair, or simply a symmetric pair.The set

$$X=X_{G}:=G^{\theta,-}$$

is called the symmetric space attached to $(G,H)$, and it is equipped with the natural action of $G$ by twisted conjugation:

$$x\cdot g:=gx\theta(g)^{-1}.$$

The map

$$g\in G\to x_{g}:=g\cdot e\in X$$

induces a homeomorphism between $G/H$ and the orbit of $e$ in $X$. For $g\in G$ and $x\in X$, we set $\mathrm{Ad}(g)(x)=gxg^{-1}$. For any $x\in X$, we can twist the involution $\theta$ by $x$:

$$\theta_{x}:=x\theta x^{-1}.$$

We will mostly be interested by the twists by elements $x_{g}$ in $G\cdot e$, in which case

$$\theta_{g}:=\theta_{x_{g}}=\mathrm{Ad}(g)\circ\theta\circ\mathrm{Ad}(g)^{-1}.$$

We observe that for $g$ in $G$, $A\subseteq G$, and $x\in X$:

$$A^{\theta_{g}}=A\cap gHg^{-1},$$

and

$$\theta_{g\cdot x}=(\theta_{x})_{g}.$$

In particular

$$G^{\theta_{g}}=gHg^{-1},$$

and if $g^{\prime}\in G$ then

$$\theta_{gg^{\prime}}=(\theta_{g^{\prime}})_{g}.$$

We will freely and tacitly use these observations. Let $M$ be a Levi subgroup of $x\in X$ is $M$-admissible if $\theta_{x}(M)=M$. The following lemma, which is [Off, Lemma 6.3], is very useful.

In [Off, Section 6] and following [LR], after fixing a $\theta$-stable maximal split torus $T_{0}$ of $G$, and $P_{0}$ a minimal parabolic subgroup containing it, Offen introduced an oriented graph the vertices of of which are the couples $(M,x)$ such that $M$ is a standard Levi subgroup of $G$, and $x\in X$ is $M$-admissible. We denote this graph by $\Gamma_{G}(\theta,T_{0},P_{0})$. Now we observe that the definition of $\Gamma_{G}(\theta,T_{0},P_{0})$ does not require that $T_{0}$ is $\theta$-stable, and that [Off, Lemma 6.4] still holds for $\Gamma_{G}(\theta,T_{0},P_{0})$ for a random choice of $T_{0}$. All that matters is that if $(M,x)$ is a vertex of $\Gamma_{G}(\theta,T_{0},P_{0})$, then $\theta_{x}(M)=M$ and hence $\theta_{x}$ acts on the roots of $A_{M}$ in $G$. We then define the graph $\Gamma_{G}(\theta)$ to be the oriented graph given by the disjoint union of the oriented graphs $\Gamma_{G}(\theta,T_{0},P_{0})$, for $T_{0}$ a maximal split torus of $G$, and $P_{0}$ a minimal parabolic subgroup of $G$ containing it. In particular a vertex could be labeled by a given pair $(M,x)$ such that $x$ is $M$-admissible more than one time, but it does not matter.

Now we want to define the notion of unimodularity for an $M$-admissible elements $x\in X$, which is referred to as the modulus assumption in [MOY].

The unimodularity assumption is satisfied in many interesting cases. We refer to [BM] for the terminology of pairs of Galois type, of PTB type, and of diagonal type (the latter corresponding to the so called group case).

It is worthwhile observing that unimodular pairs have the nice property of being automatically tempered, at least when $F$ is $p$-adic, though this should be true as well when $F$ is Archimedean. We recall that whenever $(G,H)$ is a symmetric pair, there exists a unique right $G$-invariant measure on $H\backslash G$, up to a positive scalar. With respect to such a measure, the space of functions $L^{2}(H\backslash G)$ becomes a unitary representation of $G$, and it admits a unique class of Plancherel measures.

Here we are following the terminology of [BP1, Section 2.7]. Comparing [BP1, Lemma 2.7.1] and [Zha, Section 3.2], we see that tempered pairs are exactly the pairs called very strongly discrete in [Zha]. In particular, by [Zha, Section 3.2.2, (19)], we deduce the following result.

We recall that by definition, an irreducible representation of $G$ is called square-integrable if it is unitary, and if its matrix coefficients are square-integrable mod the center of $G$. We then remind that tempered pairs are strongly discrete in the terminology of [GO], as follows from [GO, Theorem 1.1] and [Zha, Proposition 3.3], i.e.:

Let $(G,H)$ be a tempered symmetric pair, and $\pi$ be a square-integrable representation of $G$. To each $v\in\pi$, and $v^{\vee}$ in the contragredient $\pi^{\vee}$ of $\pi$, we can associate the matrix coefficient

$$c_{v,v^{\vee}}:g\to\langle\pi(g)v,v^{\vee}\rangle.$$

Then by Proposition 1.11, the linear form

$$\ell_{v^{\vee}}:v\to\int_{H/A_{G}\cap H}c_{v,v^{\vee}}(h)dh$$

is a well-defined element in $\mathrm{Hom}_{H}(\pi,{\mathbb{C}}).$ We set

$$\mathcal{H}(\pi)=\{\ell_{v^{\vee}},\ v^{\vee}\in\pi^{\vee}\}\subseteq\mathrm{Hom}_{H}(\pi,{\mathbb{C}}).$$

We now state [Zha, Theorem 1.4], which claims that all linear forms in $\mathrm{Hom}_{H}(\pi,{\mathbb{C}})$ can be canonically expressed in terms of matrix coefficients.

We will mostly be interested in representations induced from square-integrable ones. Hence the following result is useful.

Let $F$ and $G$ be as in Section 1. Let $P$ be a parabolic subgroup of $G$, and $M$ a Levi component of $P$.

Now let $\sigma$ be a (complex, smooth admissible) finite length representation of $M$. We denote by $I_{P}^{G}(\sigma)$ the representation of $G$ obtained from normalized parabolic induction of $\sigma$. We denote by $X^{*}(M)$ the lattice of algebraic characters of $M$, which is a lattice of rank $d$ which is the dimension of $A_{M}$. We set

$$\mathfrak{a}_{M}^{*}:={\mathbb{R}}\otimes_{{\mathbb{Z}}}X^{*}(M)\simeq{\mathbb{R}}^{d}$$

and

$$\mathfrak{a}_{M,{\mathbb{C}}}^{*}:={\mathbb{C}}\otimes_{{\mathbb{Z}}}X^{*}(M)=\mathfrak{a}_{M}^{*}+i\mathfrak{a}_{M}^{*}\simeq{\mathbb{C}}^{d}.$$

Hence for $\underline{s}\in\mathfrak{a}_{M,{\mathbb{C}}}^{*}$, its real part $\mathfrak{R}(\underline{s})\in\mathfrak{a}_{M}^{*}$ is well defined. Then we have a natural homomorphism

$$\Phi:\mathfrak{a}_{M,{\mathbb{C}}}^{*}\to\mathrm{Hom}(M,{\mathbb{C}}^{\times})$$

acting on pure tensors by the formula

$$\Phi(s\otimes\chi)=|\chi(m)|_{F}^{s}.$$

We denote by

$$X^{0}(M):={\mathrm{Im}\,}(\Phi)$$

the image of this map, which we call the group of unramified characters of $M$. In particular elements in $X^{0}(M)$ are naturaly parametrized by vectors $\underline{s}\in\mathfrak{a}_{M,{\mathbb{C}}}^{*}$, and more precisely we set

$$\chi_{\underline{s}}:=\Phi(\underline{s}).$$

For $\underline{s}\in\mathfrak{a}_{M}^{*}$ we set

$$\sigma[\underline{s}]:=\chi_{\underline{s}}\otimes\sigma.$$

We fix for the rest of this section a maximal compact subgroup $K$ of $G$ which is in good position with respect to $(P,M)$ in the following sense:

1. (a)

   $G=PK$.

2. (b)

   $M\cap K$ is a maximal compact subgroup of $M$.

3. (c)

   If $P=MU$ is the Levi decomposition of $P$ with respect to $M$, then $P\cap K=(M\cap K)(U\cap K).$ Note that this actually automatically follows from the second condition.

Then to $f\in I_{P}^{G}(\sigma)$, one can attach a holomorphic section

$$f_{\underline{s}}\in I_{P}^{G}(\sigma[\underline{s}])$$

where $\underline{s}$ varies in $\mathfrak{a}_{M}^{*}$. Explicitly putting

$$\eta_{\underline{s}}(umk)=\chi_{s}(m)$$

for $u\in U$, $m\in M$ and $k\in K$, then $f_{\underline{s}}=\eta_{\underline{s}}f$.

Now suppose that $(G,H)$ is a symmetric pair associated to the involution $\theta$ of $G$. Write

$$G=\coprod_{u\in R(P\backslash G/H)}PuH.$$

Throughout the paper we will always make the following assumption on $R(P\backslash G/H)$.

For any vertex $(M,x_{u})$ (where $u\in R(P\backslash G/H)$ always), we denote by $X^{0}(M)^{\theta_{u},-}$ the subgroup of $\theta_{u}$-anti-invariant characters inside $X^{0}(M)$:

$$X^{0}(M)^{\theta_{u},-}:=\{\chi\in X^{0}(M),\ \chi\circ\theta_{u}=\chi^{-1}\}.$$

The group $X^{0}(M)^{\theta_{u},-}$ is the image under the map $\Phi$ of the complex vector space

$$\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1):=\{\underline{s}\in\mathfrak{a}_{M,{\mathbb{C}}}^{*},\ \theta_{u}(\underline{s})=-\underline{s}\},$$

and we set

$$\mathfrak{a}_{M}^{*}(\theta_{u},-1):=\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1)\cap\mathfrak{a}_{M}^{*}.$$

Now suppose that the vertex $(M,x_{u})$ is unimodular with respect to $P$, and fix an invariant linear form in

$$\ell\in\mathrm{Hom}_{M^{\theta_{u}}}(\sigma,{\mathbb{C}}).$$

It is proved in [MOY] (which considers in fact any vertex $(M,x)$ unimodular with respect to $P$) that if $f_{\underline{s}}$ is a holomorphic section of $I_{P}^{G}(\sigma[\underline{s}])$, the integral

$$\int_{u^{-1}Pu\cap H\backslash H}\ell(f_{\underline{s}}(uh))dh$$

is formally well-defined because $x_{u}$ is unimodular, and convergent for $\mathfrak{R}(\underline{s})$ in $\mathcal{D}_{x_{u},\sigma}$ where $\mathcal{D}_{x_{u},\sigma}\subseteq\mathfrak{a}_{M}^{*}(\theta_{u},-1)$ is a non empty open cone independent of $f$. This defines a linear form

$$J_{x_{u},\sigma,\ell,\underline{s}}:f_{\underline{s}}\to\int_{u^{-1}Pu\cap H\backslash H}\ell(f_{\underline{s}}(uh))dh$$

in $\mathrm{Hom}_{H}(I_{P}^{G}(\sigma[\underline{s}]),{\mathbb{C}})$ whenever $\mathfrak{R}(\underline{s})\in\mathcal{D}_{x_{u},\sigma}$. It is then proved in [MOY] that this family of linear forms extends meromorphically to $\underline{s}\in\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1)$ in the following sense: for each $\underline{s}_{0}\in\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1)$, there exists a nonzero meromorphic function $F_{\underline{s}_{0}}(\underline{s})$ on $\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1)$ such that $F_{\underline{s}_{0}}(\underline{s})J_{x_{u},\sigma,\ell,\underline{s}}(f_{\underline{s}})$ is holomorphic at $\underline{s}_{0}$ for each $f\in I_{P}^{G}(\sigma)$. When $F$ is $p$-adic we can actually choose $F_{\underline{s}_{0}}$ independent of $\underline{s}$, and polynomial in the variable $(q^{\pm s_{1}},\dots,q^{\pm s_{d}})$ for $q$ the residual characteristic of $F$, where the $s_{i}$ pertain to the choice of a basis of $\mathfrak{a}_{M,{\mathbb{C}}}^{*}$. Finally if $\ell\neq 0$, then the intertwining period $J_{x_{u},\sigma,\ell,\underline{s}}(f_{\underline{s}})$ is nonzero for at least one $f\in I_{P}^{G}(\sigma)$.

From now on, we assume that $(G,H)$ is unimodular. If $\ell\neq 0\in\mathrm{Hom}_{M^{\theta_{u}}}(\sigma,{\mathbb{C}})$, then for a generic choice of $\underline{s}_{0}\in\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1)-\{0\}$, there exists $k(\underline{s}_{0})\in{\mathbb{Z}}$ such that the linear map

(2.1)

$$J_{x_{u},\sigma,\ell}^{*,\underline{s}_{0}}:=\lim_{s\to 0}s^{k(\underline{s}_{0})}J_{x_{u},\sigma,\ell,s\times\underline{s}_{0}}$$

is nonzero. More generally, let $\mathfrak{v}$ be a vector subspace of $\mathfrak{a}_{M,{\mathbb{C}}}^{*}$. We denote by

$$R_{\mathfrak{v}}(P\backslash G/H)\subseteq R(P\backslash G/H)$$

the subset of representatives such that $(M,x_{u})$ is a vertex and such that $\mathfrak{a}_{M,{\mathbb{C}}}^{*}(\theta_{u},-1)=\mathfrak{v}$. Set

$$R_{\mathfrak{v}}(P\backslash G/H)=\{u_{1}^{\mathfrak{v}},\dots,u_{k_{\mathfrak{v}}}^{\mathfrak{v}}\},$$

and for each $i=1,\dots,k_{\mathfrak{v}}$, select $\ell_{i}^{\mathfrak{v}}\in\mathrm{Hom}_{M^{\theta_{{u_{i}}^{\mathfrak{v}}}}}(\sigma,{\mathbb{C}})$. If

$$J_{\mathfrak{v},\sigma,\ell^{\mathfrak{v}},\underline{s}}:=\sum_{i=1}^{k_{\mathfrak{v}}}J_{x_{u_{i}^{\mathfrak{v}}},\sigma,\ell_{i}^{d},\underline{s}}$$

is nonzero, then for a generic choice of $\underline{s}_{0}\in\mathfrak{v}-\{0\}$, there exists $k(\underline{s}_{0})\in{\mathbb{Z}}$ such that the linear map

(2.2)

$$J_{\mathfrak{v},\sigma,\ell^{\mathfrak{v}}}^{*,\underline{s}_{0}}:=\lim_{s\to 0}s^{k(\underline{s}_{0})}J_{\mathfrak{v},\sigma,\ell^{\mathfrak{v}},s\times\underline{s}_{0}}$$

is nonzero.

Now to each vertex $(M,x_{u})$, we attach a meromorphic function $\mathfrak{n}(x_{u},\sigma,\underline{s})$ which we call a normalizing factor. Then we can define normalized intertwining periods.

We recall from [vdB] and [Del] that the space $\mathrm{Hom}_{H}(I_{P}^{G}(\sigma),{\mathbb{C}})$ is finite dimensional whenever $(G,H)$ is a symmetric pair, for $\sigma$ of finite length. The goal of this paper is to study the following question.

A positive answer to the above question can be useful. For example in [Mat4] and [SX] it is used to study distinction of quotients of induced representations, such as discrete series or Speh representations, whereas in [ALM⁺] it is used to compute the sign of linear periods. We hope that the answer is yes in the following case. We recall that $\sigma$ is called square-integrable if it is unitary, and its matrix coefficients are square-integrable mod the center of $M$.

In the following remark, we discuss a possible extension of the conjecture to more general pairs.

In this section $(G,H)$ is unimodular. We check that the basic properties of intertwining periods (see Remark 2.2), as well the statement of Conjecture A are independent of some choices. For example, in Section 2, we used a fixed maximal compact subgroup of $G$ in good position with respect to $(P,M)$ to define holomorphic sections. Here we observe that if Conjecture A holds for one choice of maximal compact subgroup well positioned with respect to $(P,M)$, then it holds for any. Then we also prove that Conjecture A does not depend on the conjugacy class of $H$ inside $G$, in the sense that if it true for $H$, it is automatically true for $gHg^{-1}$ whenever $g\in G$. Then we check a generalization of this property, which for example applies when $G$ is a special linear group but $g$ above belongs to the corresponding general linear group. The reason in each case is a similar argument.