Low degree cohomologies of congruence groups

TL;DR

This paper investigates the low degree cohomologies of congruence groups, proving their vanishing in some cases and explicitly determining them in others, advancing understanding in algebraic and number theoretic contexts.

Contribution

It establishes new vanishing results and explicit calculations for low degree cohomologies of congruence groups, extending prior knowledge in the field.

Findings

Vanishing of certain low degree cohomologies of induced representations.

Explicit determination of low degree cohomologies of some congruence groups.

Applications to algebraic and number theory contexts.

Abstract

We prove the vanishing of certain low degree cohomologies of some induced representations. As an application, we determine certain low degree cohomologies of congruence groups.

Tables

Table 1. Table 1. Complex simple Lie algebras

$𝔤$	$r_{𝔤}^{\prime }$	$r_{𝔤}$
$𝔰{𝔩}_n\left(ℂ\right)\left(n\ge 2\right)$	$2\left(n-1\right)$	$n-1$
$𝔰{𝔬}_n\left(ℂ\right)\left(n\ge 7\right)$	$2\left(n-2\right)$	$n-2$
$𝔰{𝔭}_{2n}\left(ℂ\right)\left(n\ge 2\right)$	$2\left(2n-1\right)$	$2n-1$
${𝔢}_6\left(ℂ\right)$	$32$	$16$
${𝔢}_7\left(ℂ\right)$	$54$	$27$
${𝔢}_8\left(ℂ\right)$	$114$	$57$
${𝔣}_4\left(ℂ\right)$	$30$	$15$
${𝔤}_2\left(ℂ\right)$	$10$	$5$

Table 2. Table 2. Noncompact noncomplex simple classical Lie algebras

$𝔤$	$r_{𝔤}^{\prime }$	$r_{𝔤}$
$𝔰{𝔩}_n\left(ℝ\right)\left(n\ge 2\right)$	$n-1$	$n-1$
$𝔰{𝔭}_{2n}\left(ℝ\right)\left(n\ge 3\right)$	$2n-1$	$n$
$𝔰𝔬(p,q)(p\ge q\ge 1,p+q\ge 5,(p,q)\ne (3,3))$	$p+q-2$	$q$
$𝔰𝔲(p,q)(p\ge q\ge 1,p+q\ge 3,(p,q)\ne (2,2))$	$2\left(p+q\right)-3$	$q$
$𝔰𝔭(p,q)(p\ge q\ge 1,p+q\ge 3,(p,q)\ne (2,2))$	$4\left(p+q\right)-5$	$2q$
$𝔰{𝔩}_n\left(ℍ\right)\left(n\ge 4\right)$	$4\left(n-1\right)$	$2\left(n-1\right)$
$𝔰{𝔬}^{\ast }\left(2n\right)\left(n=5,7,8,9,\cdots \right)$	$4n-7$	$n-1$
$𝔰{𝔩}_3\left(ℍ\right)$	$8$	$3$
$𝔰{𝔬}^{\ast }\left(12\right)$	$15$	$5$
$𝔰𝔭(2,2)$	$10$	$4$

Table 3. Table 3. Noncompact noncomplex exceptional simple Lie algebras

$𝔤$	$r_{𝔤}^{\prime }$	$r_{𝔤}$
E I	$16$	$11$
E II	$21$	$8$
E III	$21$	$6$
E IV	$16$	$6$
E V	$27$	$15$
E VI	$33$	$12$
E VII	$27$	$11$
E VIII	$57$	$29$
E IX	$57$	$24$
F I	$15$	$8$
F II	$15$	$4$
G	$5$	$3$

Equations

$$H_{\mathrm{ct}}^{r_{G,F}}(G;F\otimes \pi )\ne \{0\}$$

$$r_G:=\min \{r_{G,F}:F\text{ is an irreducible finite-dimensional representation of G}\}.$$

$$q_0(G):=\frac{\dim G/K-(\mathrm{rank}{\mathfrak{g}}_{\mathbb{C}}-\mathrm{rank}{\mathfrak{k}}_{\mathbb{C}})}{2},$$

$$⟨\nu {|}_{{\mathfrak{h}}_{\mathbb{C}}},{\alpha }^{\vee }⟩\ge 0\text{for all }\alpha \in \Delta ({\mathfrak{h}}_{\mathbb{C}},{\mathfrak{n}}_{\mathbb{C}}).$$

$$H_{\mathrm{ct}}^i(G;F\otimes {\mathrm{Ind}}_P^G(\nu \otimes \sigma ))=\{0\}\text{for all }i<r_G.$$

$$H_{\mathrm{ct}}^i(G;F\otimes {\mathrm{Ind}}_P^G(\nu \otimes \sigma ))=\{0\}$$

$$H_{\mathrm{ct}}^{q_{G,F}}(G;F\otimes {\mathrm{Ind}}_P^G(\nu \otimes \sigma ))\ne \{0\}$$

$$q_G:=\min \{q_{G,F}:F\text{ is a finite-dimensional irreducible representations of G}\}.$$

$$r_G\le q_G.$$

$$G=A_{\mathsf{G}}\times {}^{\circ }G,$$

$${}^{\circ }G:=\bigcap _{\chi \text{ is an algebraic character of G defined over Q}}\ker (|\chi |:\mathsf{G}(\mathbb{R})\to {\mathbb{R}}_{+}^{\times }).$$

$$F^{\Gamma }\otimes H_{\mathrm{ct}}^i({}^{\circ }G;\mathbb{C})=H_{\mathrm{ct}}^i({}^{\circ }G;F^{\Gamma })\to H^i(\Gamma ;F),(i\in \mathbb{Z}).$$

$$H_{\mathrm{ct}}^i({}^{\circ }G;\mathbb{C})={\mathrm{Hom}}_K({\wedge }^i({}^{\circ }{\mathfrak{g}}_{\mathbb{C}}/{\mathfrak{k}}_{\mathbb{C}}),\mathbb{C}),$$

$$H^i(\Gamma ;\mathbb{C})=H_{\mathrm{ct}}^i({}^{\circ }G;\mathbb{C})\text{ for all i≤⌊4n+2⌋}.$$

$$H^i(\Gamma ;F)=F^{\Gamma }\otimes H_{\mathrm{ct}}^i({}^{\circ }G;\mathbb{C})\text{ for all i≤n−2},$$

$$V\to V,v\mapsto X.v:=\frac{d}{dt}{|}_{t=0}\exp (tX).v$$

$$H_{\mathrm{ct}}^i(G;V)=H_{\mathrm{sm}}^i(G;V),(i\in \mathbb{Z})$$

$$H_{\mathrm{sm}}^i(G;V)=H^i({\mathfrak{g}}_{\mathbb{C}},K;V),(i\in \mathbb{Z})$$

$$V^{\mathfrak{J}}:=\{v\in V:X.v=0,\text{for all }X\in \mathfrak{J}\},$$

$$\underset{\mathfrak{J}}{\underrightarrow{\lim }}{V}^{\mathfrak{J}}\to V$$

$$P=\mathsf{P}(\mathbb{R}),N_{\mathsf{P}}:={\mathsf{N}}_{\mathsf{P}}(\mathbb{R})\text{and}{L}_{\mathsf{P}}:={\mathsf{L}}_{\mathsf{P}}(\mathbb{R}).$$

$$P\to L_{\mathsf{P}}.$$

$$|(X.\varphi )(g)|\le C_X\cdot |g{|}_{\varphi },\text{for all }g\in G.$$

$$|\varphi {|}_X:=\underset{g\in G}{\sup }\frac{|(X.\varphi )(g)|}{|g{|}_{\mathfrak{J}}}<\infty ,$$

$$\mathcal{A}({\Gamma }_{\mathsf{P}}{N}_{\mathsf{P}}\backslash G)=\underset{\mathfrak{J}}{\underrightarrow{\lim }}(\mathcal{A}({\Gamma }_{\mathsf{P}}{N}_{\mathsf{P}}\backslash G){)}^{\mathfrak{J}}.$$

$$V^{K_f,\mathfrak{J}}:=\{v\in V:k.v=v,X.v=0,\text{for all }k\in K_f,X\in \mathfrak{J}\}$$

$$\underset{K_f,\mathfrak{J}}{\underrightarrow{\lim }}{V}^{K_f,\mathfrak{J}}\to V$$

$$\mathsf{P}(\mathbb{A})\to {\mathsf{L}}_{\mathsf{P}}(\mathbb{A}).$$

$$\begin{array}[]{rcl}\Gamma_{\mathsf{P},g_{f}K_{f}}N_{\mathsf{P}}\backslash G&\rightarrow&(\mathsf{L}_{\mathsf{P}}(\mathbb{Q})\mathsf{N}_{\mathsf{P}}(\mathbb{A}))\backslash\mathsf{G}(\mathbb{A})/K_{f},\\ \textrm{the coset of $g\in G$}&\mapsto&\textrm{the double coset of $(g,g_{f})\in\mathsf{G}(\mathbb{A})$}\\ \end{array}$$

$$\mathsf{P}({\mathbb{A}}_f)\to {\mathsf{L}}_{\mathsf{P}}({\mathbb{A}}_f).$$

Full text

1. Introduction

1.1. Vanishing of low degree cohomologies of some induced representations

Let $G$ be a real reductive group, namely, it is a Lie group with the following properties:

1. $\bullet$

its Lie algebra $\mathfrak{g}$ is reductive; 2. $\bullet$

$G$ has only finitely many connected components; 3. $\bullet$

there is a connected closed subgroup of $G$ with finite center whose Lie algebra equals $[\mathfrak{g},\mathfrak{g}]$.

Here and henceforth, unless otherwise specified, we use the corresponding lowercase Gothic letter to indicate the Lie algebra of a Lie group. We use a subscript $\mathbb{C}$ to indicate the complexification of a real vector space. For example, $\mathfrak{g}_{\mathbb{C}}$ is the complexified Lie algebra of $G$.

Let $F$ be an irreducible finite-dimensional representation of $G$. Define $r_{G,F}$ to be the smallest integer, if it exists, such that the continuous group cohomology

[TABLE]

for some infinite-dimensional irreducible unitary representations $\pi$ of $G$. If such an integer does not exist, we set $r_{G,F}=\infty$. See [HM, Section 2] for the definition of continuous cohomologies.

Define

[TABLE]

The quantity $r_{G}$ is calculated in [En, Table 1], [Ku, Theorem 2, Theorem 3], [VZ, Table 8.2] and [LS2, Section 2.B]. See also Tables 1, 2 and 3 of Section 5.4.

Let $K$ be a maximal compact subgroup of $G$.

Examples 1.1*.*

(a). If the Lie algebra $\mathfrak{g}$ has no non-compact simple ideal, then $G$ has no infinite-dimensional irreducible unitary representation. Thus in this case $r_{G,F}=\infty$ for all $F$, and $r_{G}=\infty$.

(b). If all highest weights of $F$ are regular, then $r_{G,F}\geq q_{0}(G)$, where

[TABLE]

and “${\mathrm{rank}}$” stands for the rank of a reductive complex Lie algebra. See [LS1, Proposition 4.2 and Proposition 4.4].

(c) If $G={\mathrm{GL}}_{n}(\mathbb{R})$ ($n\geq 2$), then $r_{G}=n-1$.

Let $P$ be a parabolic subgroup of $G$, namely the normalizer in $G$ of a parabolic subalgebra of $\mathfrak{g}$. Denote by $N$ its unipotent radical and write $L:=P/N$ for its Levi quotient.

Definition 1.2**.**

A positive character $\nu:L\rightarrow\mathbb{R}^{\times}_{+}$ is said to be dominant if

[TABLE]

Here the compleixfied differential of $\nu:L\rightarrow\mathbb{R}^{\times}_{+}$ is denoted by $\nu:\mathfrak{l}_{\mathbb{C}}\rightarrow\mathbb{C}$, $\mathfrak{h}$ is a maximally split Cartan subalgebra of $\mathfrak{l}$, and a splitting $L\rightarrow P$ is fixed so that $\mathfrak{h}$ is viewed as a Cartan subalgebra of $\mathfrak{g}$.

Here and henceforth, $\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{n}_{\mathbb{C}})$ denotes the subset of the root system $\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ consisting of the roots attached to $\mathfrak{n}_{\mathbb{C}}$, and $\alpha^{\vee}$ denotes the coroot corresponding to the root $\alpha$. Note that $\langle\nu|_{\mathfrak{h}_{\mathbb{C}}},\alpha^{\vee}\rangle$ is a real number as $\nu$ is a positive character of $L$.

Remark 1.3*.*

A splitting $L\rightarrow P$ exists and is unique up to the conjugation by a unique element of $N$. This implies that Definition 1.2 is independent of the choices of $\mathfrak{h}$ and the splitting $L\rightarrow P$.

The following theorem is the main local result of this paper.

Theorem 1.4**.**

Suppose that $P$ is a proper parabolic subgroup of $G$. Let $\nu:L\rightarrow\mathbb{R}^{\times}_{+}$ be a dominant positive character. Then for every irreducible unitarizable Casselman-Wallach representation $\sigma$ of $L$, and every irreducible finite-dimensional representation $F$ of $G$,

[TABLE]

In this article, “${\mathrm{Ind}}$” always stands for the normalized smooth induction. Recall that a representation of a real reductive group is called a Casselman-Wallach representation if it smooth, Fréchet, of moderate growth, and its Harish-Chandra module has finite length. The reader is referred to [Ca], [Wa2, Chapter 11] or [BK] for details about Casselman-Wallach representations.

Remark 1.5*.*

In Theorem 1.4, if we further assume that all the highest weights of $F$ are regular, and $\sigma$ is tempered, then it is proved in [LS1, Theorem 4.5] that

[TABLE]

for all $i<q_{0}(G)$ (see (1)).

Similar to $r_{G,F}$, we define $q_{G,F}$ to be the smallest integer, if it exists, such that the continuous group cohomology

[TABLE]

for some proper parabolic subgroups $P$ of $G$, some unitarizable irreducible Casselman-Wallach representations $\sigma$ of $L$, and some dominant positive characters $\nu$ of $L$. If no such integer exist, we set $q_{G,F}=\infty$. Define

[TABLE]

Theorem 1.4 amounts to saying that

[TABLE]

1.2. Low degree cohomologies of congruenc groups

Now we further assume that $G=\mathsf{G}(\mathbb{R})$, where $\mathsf{G}$ is a connected reductive linear algebraic group defined over ${\mathbb{Q}}$. Let $\Gamma$ be a congruence subgroup of $\mathsf{G}({\mathbb{Q}})$, namely, $\Gamma$ has a finite index subgroup of the form $\mathsf{G}({\mathbb{Q}})\cap K_{f}$, where $K_{f}$ is an open compact subgroup of $\mathsf{G}(\mathbb{A}_{f})$. Here $\mathbb{A}_{f}$ denotes the ring of finite adeles of ${\mathbb{Q}}$, and $\mathsf{G}({\mathbb{Q}})$ is viewed as a subgroup of $G$, as well as a subgroup of $\mathsf{G}(\mathbb{A}_{f})$.

Recall that $F$ is an irreducible finite dimensional representation of $G$. The group cohomology (also named the Eilenberg-MacLane cohomology) $\operatorname{H}^{i}(\Gamma;F)$ is of great interest in both geometry and arithmetic. If we viewed $\Gamma$ as a discrete group, then $\operatorname{H}^{i}(\Gamma;F)$ equals the continuous cohomology $\operatorname{H}_{\mathrm{ct}}^{i}(\Gamma;F)$.

Let $\mathsf{A}_{\mathsf{G}}$ denote the largest central split torus in $\mathsf{G}$, and write $A_{\mathsf{G}}$ for the identity connected component of $\mathsf{A}_{\mathsf{G}}(\mathbb{R})$. Then we have a decomposition

[TABLE]

where

[TABLE]

Note that $\Gamma\subset{{}^{\circ}\!G}$. By restriction of continuous cohomology, the embedding $F^{\Gamma}\hookrightarrow F$ induces a linear map

[TABLE]

Here and henceforth, a superscript group indicates the invariant vectors under the group action, and here we let both $\mathbb{C}$ and $F^{\Gamma}$ carry the trivial representation of ${{}^{\circ}\!G}$.

Remark 1.6*.*

Note that $K\subset{{}^{\circ}\!G}$, and

[TABLE]

where the last $\mathbb{C}$ stands for the trivial representation of $K$. See [BW, Chapter II, Corollary 3.2] or [Wa1, Proposition 9.4.3].

At least when $F$ is the trivial representation, there are quite a few works which study the surjectivity and the injectivity of the map (4). See [Ga, GH, Bo, Ya], for examples. The following theorem heavily relies on the work of Franke [Fr, Sections 7.4 and 7.5].

Theorem 1.7**.**

If $i\leq q_{G,F}$, then the map (4) is injective. If $i<\min\{r_{G,F},q_{G,F}\}$, then the map (4) is a linear isomorphism.

Combining Theorem 1.4 and Theorem 1.7, we get the following result.

Theorem 1.8**.**

If $i<r_{G}$, then the map (4) is a linear isomorphism.

Remarks 1.9*.*

(a) When all the highest weights of $F$ are regular, it is proved by Li-Schwermer in [LS1, Corollary 5.6] that $\operatorname{H}^{i}(\Gamma;F)=\{0\}$ when $i<q_{0}(G)$ (see (1)).

(b) Assume that $F$ is the trivial representation. It was proved by Borel in [Bo, Theorem 7.5] that the map (4) is injective when $i\leq c(\mathsf{G})$, where $c(\mathsf{G})$ is a certain constant attached to $\mathsf{G}$ (see [Bo, Section 9]). It was later proved by Yang in [Ya] that the map (4) is injective when $i\leq 2c(\mathsf{G})+1$. Based on the works of Garland-Hsiang [GH] and Garland [Ga], Borel also proved in [Bo, Theorem 7.5] that the map (4) is a linear isomorphism when $i\leq\min\{c(\mathsf{G}),m(G)\}$, where $m(G)$ is a certain constant attached to the real group $G$. In the works of Borel and Yang, $\Gamma$ is allowed to be any arithmetic group.

Example 1.10*.*

Suppose $\mathsf{G}={\mathrm{GL}}(n)/_{{\mathbb{Q}}}$ ($n\geq 2$). Then $c(\mathsf{G})=\lfloor\frac{n}{2}\rfloor-1$ (see [Bo, Section 9.2]) and $m(G)=\frac{n+2}{4}$ (see [KN, Theorem 4.1]). Suppose $n\geq 6$ so that $\frac{n+2}{4}\leq\lfloor\frac{n}{2}\rfloor-1$. Borel’s result implies that

[TABLE]

It was pointed out by Franke [Fr, Section 7.5] that it would be interesting to get an improvement of Borel’s bound for certain groups. Theorem 1.8 implies that

[TABLE]

and for all irreducible finite dimensional representations $F$ of $G$. Thus Theorem 1.8 does improve Borel’s bound.

2. Preliminaries on representations and their cohomologies

2.1. Continuous cohomologies

Let $G$ be a locally compact Hausdorff topological group. By a representation of $G$, we mean a quasi-complete Hausdorff locally convex topological vector space $V$ over $\mathbb{C}$, together with a continuous linear action of $G$ on it.

Using the $G$-intertwining continuous linear maps as morphisms, all representations of $G$ clearly form a category. Using the strong injective resolutions (or called continuously injective resolutions) in this category, one defines the continuous cohomology group $\operatorname{H}^{i}_{\mathrm{ct}}(G;V)$ for every representation $V$ of $G$. This is a locally convex topological vector space over $\mathbb{C}$ which may or may not be Hausdorff. See [HM, Section 2] or [Bl] for details.

2.2. Smooth cohomologies

Now we suppose that $G$ is a Lie group. We say that a representation $V$ of $G$ is smooth if for every $X\in\mathfrak{g}$, the map

[TABLE]

is well-defined and continuous. When this is the case, by using (5), $V$ is naturally a representation of $\mathfrak{g}_{\mathbb{C}}$.

Using the $G$-intertwining continuous linear maps as morphisms, all smooth representations of $G$ also form a category. Using the strong injective resolutions (or called differentiably injective resolutions) in this category, we define the smooth cohomology group $\operatorname{H}^{i}_{\mathrm{sm}}(G;V)$ for every smooth representation $V$ of $G$. This is also a locally convex topological vector space over $\mathbb{C}$ which may or may not be Hausdorff. See [HM, Section 5] for details.

The first assertion of the following theorem is proved in [HM, Theorem 5.1], and the second one follows from [HM, Theorem 6.1].

Theorem 2.1**.**

Let $V$ be a smooth representation of $G$. Then there is an identification

[TABLE]

of topological vector spaces. If $G$ has only finitely many connected components, then

[TABLE]

as topological vector spaces, where $K$ is a maximal compact subgroup of $G$.

Here $\operatorname{H}^{i}(\mathfrak{g}_{\mathbb{C}},K;V)$ denotes the relative Lie algebra cohomology. See [KV, (2.127)] for the explicit complex which computes the relative Lie algebra cohomology.

2.3. Quasi Casselman-Wallach representations

Now suppose that $G$ is a real reductive group. Let $\operatorname{U}(\mathfrak{g}_{\mathbb{C}})$ denote the universal enveloping algebra of $\mathfrak{g}_{\mathbb{C}}$, whose center is denoted by $\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})$.

Definition 2.2**.**

A smooth representation $V$ of $G$ is called a quasi Casselman-Wallach representation if

• •

for every ideal $\mathfrak{J}$ of $\operatorname{U}(\mathfrak{g}_{\mathbb{C}})^{G}$ of finite co-dimension, the space

[TABLE]

which is $G$-stalbe, is a Casselman-Wallach representation of $G$ under the subspace topology;

• •

in the category of complex locally convex topological vector spaces, the natural map

[TABLE]

is a topological linear isomorphism.

We will be concerned with the continuous cohomologies of some quasi Casselman-Wallach representations.

3. Preliminaries on automorphic forms

3.1. Spaces of automorphic forms

We now return to the notation of the Introduction. In particular, $\mathsf{G}$ is a connected reductive linear algebraic group over ${\mathbb{Q}}$, and $G=\mathsf{G}(\mathbb{R})$.

Recall that a subgroup $\Gamma$ of $\mathsf{G}({\mathbb{Q}})$ is called an arithmetic subgroup if it is commensurable to the subgroup $\eta^{-1}({\mathrm{GL}}_{k}(\mathbb{Z}))\subset\mathsf{G}({\mathbb{Q}})$, where $k\geq 0$ and $\eta:\mathsf{G}\rightarrow({\mathrm{GL}}_{k})/_{{\mathbb{Q}}}$ is an embedding of algebraic groups over ${\mathbb{Q}}$. This definition is independent of $k$ and $\eta$. All congruence subgroups of $\mathsf{G}({\mathbb{Q}})$ are arithmetic subgroups.

Let $\mathsf{P}$ be a parabolic subgroup of $\mathsf{G}$. Denote by $\mathsf{N}_{\mathsf{P}}$ its unipotent radical, and write $\mathsf{L}_{\mathsf{P}}:=\mathsf{P}/\mathsf{N}_{\mathsf{P}}$ for its Levi quotient. Then $\mathsf{L}_{\mathsf{P}}$ is also a connected reductive linear algebraic group over ${\mathbb{Q}}$. Write

[TABLE]

Let $\Gamma_{\mathsf{P}}$ be an arithmetic subgroup of $\mathsf{L}_{\mathsf{P}}({\mathbb{Q}})$. By abuse of notation, we use $\Gamma_{\mathsf{P}}N_{\mathsf{P}}$ to denote the pre-image of $\Gamma_{\mathsf{P}}$ under the quotient map

[TABLE]

Let ${\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G)$ denote the space of smooth automorphic forms on $\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G$. Recall that a complex valued smooth function $\phi$ on $\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G$ is called a smooth automorphic form if

• •

$\phi$ is $\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})$-finite, namely the space $\{X.\phi\,:\,X\in\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})\}$ is finite dimensional (the algebra $\operatorname{U}(\mathfrak{g}_{\mathbb{C}})$ acts on the space of smooth functions by the differential of the right translations);

• •

$\phi$ is uniformly of moderate growth, namely, there exists a positive valued real algebraic function $\lvert\,\cdot\,\rvert_{\phi}$ on $G$ with the following property: for every $X\in\operatorname{U}(\mathfrak{g}_{\mathbb{C}})$, there exists $C_{X}>0$ such that

[TABLE]

For every ideal $\mathfrak{J}$ of $\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})$ of finite codimension, we have a space $({\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G))^{\mathfrak{J}}$, as defined in (6). It carries the action of $G$ by right translations. It also carries a natural Fréchet topology and is naturally a Casselman-Wallach representation of $G$, as explained in what follows. Harish-Chandra’s finiteness theorem (see [HC, Chapter I, §2, Theorem 1]) implies that the space of $K$-finite vectors in $({\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G))^{\mathfrak{J}}$ forms a finitely generated admissible $(\mathfrak{g}_{\mathbb{C}},K)$-module. Here and as before, $K$ is a maximal compact subgroup of $G$. Using this theorem and Casselman-Wallach’s smooth globalization theorem (see [Ca] or [Wa2, Corollary 11.6.8]), one shows that there is a positive valued real algebraic function $\lvert\,\cdot\,\rvert_{\mathfrak{J}}$ on $G$ such that

[TABLE]

for all $\phi\in({\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G))^{\mathfrak{J}}$ and $X\in\operatorname{U}(\mathfrak{g}_{\mathbb{C}})$. Under the seminorms $\{\lvert\,\cdot\,\rvert_{X}\}_{X\in\operatorname{U}(\mathfrak{g}_{\mathbb{C}})}$, $({\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G))^{\mathfrak{J}}$ becomes a Fréchet space, and is a Casselman-Wallach representation of $G$. Moreover, the topology on $({\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G))^{\mathfrak{J}}$ is independent of the function $\lvert\,\cdot\,\rvert_{\mathfrak{J}}$ on $G$.

We equip the space ${\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G)$ with the inductive topology in the category of complex locally convex topological spaces:

[TABLE]

Then ${\mathcal{A}}(\Gamma_{\mathsf{P}}N_{\mathsf{P}}\backslash G)$ is a quasi Casselman-Walach representation of $G$.

3.2. The adelic setting

Let $\mathbb{A}=\mathbb{R}\times\mathbb{A}_{f}$ denote the ring of adeles of ${\mathbb{Q}}$. Then $\mathsf{G}(\mathbb{A})=G\times\mathsf{G}(\mathbb{A}_{f})$. A representation $V$ of $\mathsf{G}(\mathbb{A})$ is said to be smooth if

• •

it is smooth as a representation of $G$; and

• •

every vector in $V$ is fixed by an open compact subgroup of $\mathsf{G}(\mathbb{A}_{f})$.

We define a quasi Casselman-Wallach representation of $\mathsf{G}(\mathbb{A})$ to be a smooth representation $V$ of $\mathsf{G}(\mathbb{A})$ with the following properties:

• •

for every open compact subgroup $K_{f}$ of $\mathsf{G}(\mathbb{A}_{f})$, and every ideal $\mathfrak{J}$ of $\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})$ of finite co-dimension, the space (with the subspace topology)

[TABLE]

is a Casselman-Wallach representatiomation of $G$;

• •

in the category of complex locally convex topological vector spaces, the natural map

[TABLE]

is a topological linear isomorphism.

By abuse of notation, we use $\mathsf{L}_{\mathsf{P}}(\mathbb{Q})\mathsf{N}_{\mathsf{P}}(\mathbb{A})$ to denote the pre-image of $\mathsf{L}_{\mathsf{P}}(\mathbb{Q})$ under the quotient map

[TABLE]

Let $\mathcal{A}_{\mathsf{P}}(\mathsf{G})$ denote the space of smooth automorphic forms on $(\mathsf{L}_{\mathsf{P}}(\mathbb{Q})\mathsf{N}_{\mathsf{P}}(\mathbb{A}))\backslash\mathsf{G}(\mathbb{A})$. Recall that a smooth function $\phi$ on $(\mathsf{L}_{\mathsf{P}}(\mathbb{Q})\mathsf{N}_{\mathsf{P}}(\mathbb{A}))\backslash\mathsf{G}(\mathbb{A})$ is called a smooth automorphic form if

• •

$\phi$ is right invariant under some open compact subgroup $K_{f}$ of $\mathsf{G}(\mathbb{A}_{f})$;

• •

for every $g_{f}\in\mathsf{G}(\mathbb{A}_{f})$, the pullback of $\phi$ through the map

[TABLE]

is a smooth automorphic form on $(\Gamma_{\mathsf{P},g_{f}K_{f}}N_{\mathsf{P}})\backslash G$, where $\Gamma_{\mathsf{P},g_{f}K_{f}}$ denotes the intersection of $\mathsf{L}_{\mathsf{P}}({\mathbb{Q}})$ with the image $g_{f}K_{f}g_{f}^{-1}\cap\mathsf{P}(\mathbb{A}_{f})$ under the quotient map

[TABLE]

By the discussion of Section 3.1, under the right translations, the space $(\mathcal{A}_{\mathsf{P}}(\mathsf{G}))^{K_{f},\mathfrak{J}}$ (see (9)) is a Casselman-Wallach representation of $G$. Then using the inductive topology in the category of locally convex topological vector space,

[TABLE]

is a quasi Casselman-Wallach representation of $\mathsf{G}(\mathbb{A})$, under the right translations.

3.3. A decomposition of $\mathcal{A}_{\mathsf{P}}(\mathsf{G})$

Write $\mathsf{A}_{\mathsf{P}}$ for the largest central split torus in $\mathsf{L}_{\mathsf{P}}$. We have a decomposition

[TABLE]

where

[TABLE]

and $A_{\mathsf{P}}$ denotes the identity connected component of $\mathsf{A}_{\mathsf{P}}(\mathbb{R})$. Write $\mathfrak{a}_{\mathsf{P}}$ for the Lie algebra of $A_{\mathsf{P}}$.

The representation $\mathcal{A}_{\mathsf{P}}(\mathsf{G})$ also carries a locally finite linear action of $A_{\mathsf{P}}$ which commutes with the $\mathsf{G}(\mathbb{A})$-action:

[TABLE]

where $\rho_{\mathsf{P}}\in\breve{\mathfrak{a}}_{\mathsf{P}}$ denotes the half sum of the weights (with the multiplicities) associated to $\mathsf{N}_{\mathsf{P}}$. Here and henceforth, “$\,\breve{\ }\,$” indicates the dual space. The action (11) differentiates to a locally finite linear action of the commutative Lie algebra $\mathfrak{a}_{\mathsf{P},\mathbb{C}}:=\mathfrak{a}_{\mathsf{P}}\otimes_{\mathbb{R}}\mathbb{C}$, and thus we have a generalized eigenspace decomposition

[TABLE]

For each $\phi\in\mathcal{A}_{\mathsf{P}}(\mathsf{G})$ and $\lambda\in\breve{\mathfrak{a}}_{\mathsf{P},\mathbb{C}}$, write $\phi_{\lambda}$ for the ${\mathcal{A}}_{\mathsf{P}}(\mathsf{G})_{\lambda}$-component of $\phi$ with respect to the decomposition (12). The decomposition (12) is $\mathsf{G}(\mathbb{A})$-invariant. When $\mathsf{P}=\mathsf{G}$, we write ${\mathcal{A}}(\mathsf{G})$ for ${\mathcal{A}}_{\mathsf{G}}(\mathsf{G})$, and the decomposition (12) is specified to

[TABLE]

In general, the map

[TABLE]

establishes an isomorphism

[TABLE]

of Quasi Casselman-Wallach representations of $\mathsf{G}(\mathbb{A})$. Here $h^{-\rho_{\mathsf{P}}}:=a_{h}^{-\rho_{\mathsf{P}}}$, with $a_{h}$ denotes the projection of $h$ to $A_{\mathsf{P}}$ with respect to the decomposition (10).

3.4. A lemma of Langlands

Fix a pair $(\mathsf{P}_{0},\mathsf{A}_{0})$ such that $\mathsf{P}_{0}$ is a minimal parabolic subgroup of $\mathsf{G}$, and $\mathsf{A}_{0}$ is a maximal split torus in $\mathsf{P}_{0}$. Let $\mathscr{P}$ denote the set of standard parabolic subgroups of $\mathsf{G}$, namely, parabolic subgroups containing $\mathsf{P}_{0}$. Suppose that $\mathsf{P}\in\mathscr{P}$. View $\mathsf{L}_{\mathsf{P}}$ as an algebraic subgroup of $\mathsf{P}$ containing $\mathsf{A}_{0}$ so that

[TABLE]

For simplicity, write $A_{0}:=A_{\mathsf{P}_{0}}$, $\mathfrak{a}_{0}:=\mathfrak{a}_{\mathsf{P}_{0}}$. Similar abbreviations will be used without further explanation.

Denote by $\Delta_{0}^{+}\subset\breve{\mathfrak{a}}_{0}$ the positive root system attached to the pair $(\mathsf{P}_{0},\mathsf{A}_{0})$. More generally, write $\Delta_{\mathsf{P}}^{+}\subset\Delta_{0}^{+}$ for the set of roots associated to $\mathsf{N}_{\mathsf{P}}$. Given $\mathsf{P}^{\prime}\in\mathscr{P}$ with $\mathsf{P}^{\prime}\subset\mathsf{P}$, we have a decomposition

[TABLE]

where

[TABLE]

Here and as before, for each $\alpha\in\Delta_{0}^{+}$, $\alpha^{\vee}\in\mathfrak{a}_{0}$ denotes the corresponding coroot. In particular, we have

[TABLE]

Define

[TABLE]

Let ${}^{+}\!\breve{\mathfrak{a}}_{\mathsf{P}^{\prime}}^{\mathsf{P}}$ denote the interior of $\overline{{}^{+}\!\breve{\mathfrak{a}}_{\mathsf{P}^{\prime}}^{\mathsf{P}}}$ in $\breve{\mathfrak{a}}_{\mathsf{P}^{\prime}}^{\mathsf{P}}$. Define

[TABLE]

and

[TABLE]

Then ${\breve{\mathfrak{a}}}_{\mathsf{P}}^{+}$ equals the interior of $\overline{{\breve{\mathfrak{a}}}_{\mathsf{P}}^{+}}$ in ${\breve{\mathfrak{a}}}_{\mathsf{P}}$.

Note that

[TABLE]

More generally, recall the following lemma which is due to Langlands (see [BW, Lemma IV.6.11] or [Wa1, Section 5.A.1]).

Lemma 3.1**.**

There is a decomposition

[TABLE]

The decomposition (15) yields a map

[TABLE]

specified by requiring that $\nu_{+}\in\breve{\mathfrak{a}}_{\mathsf{P}}^{+}$ and $\nu\in\nu_{+}-\overline{\,{}^{+}\!\breve{\mathfrak{a}}_{0}^{\mathsf{P}}}$, for some $\mathsf{P}\in\mathscr{P}$.

3.5. Almost square integrable automorphic forms

Put

[TABLE]

We use “$\mathrm{Re}$” to indicates the real part in various context. Recall that an element $\phi$ of ${\mathcal{A}}_{\mathsf{P}}(\mathsf{G})$ belongs to ${\mathcal{A}}^{2}_{\mathsf{P}}(\mathsf{G})$ if and only if (see [MW, Lemma I.4.11])

[TABLE]

where $\phi_{\mathsf{P}^{\prime}}$ denotes the constant term of $\phi$ along $\mathsf{P}^{\prime}$, namely

[TABLE]

for all $g\in\mathsf{G}(\mathbb{A})$. Here $\operatorname{d}\!n$ is the invariant measure with total volume $1$.

As a variation of ${\mathcal{A}}^{2}_{\mathsf{P}}(\mathsf{G})$, we define

[TABLE]

and call it the space of almost square integrable automorphic forms.

It is clear that both ${\mathcal{A}}^{2}_{\mathsf{P}}(\mathsf{G})$ and ${\mathcal{A}}^{\bar{2}}_{\mathsf{P}}(\mathsf{G})$ are are closed subspaces of ${\mathcal{A}}_{\mathsf{P}}(\mathsf{G})$. Moreover, they are $\mathsf{G}(\mathbb{A})\times A_{\mathsf{P}}$-subrepresentations.

The decomposition (12) induces generalized eigenspace decompositions

[TABLE]

and the isomorphism (14) induces isomorphisms

[TABLE]

for all $\lambda\in\breve{\mathfrak{a}}_{\mathsf{P},\mathbb{C}}$.

In what follows we describe the representation ${\mathcal{A}}^{\bar{2}}(\mathsf{G})$ in terms of square integrable automorphic forms of the Levi factors. See [Fr, Section 6] for more details. View $\mathscr{P}$ as a category so that a morphism from $\mathsf{P}^{\prime}\in\mathscr{P}$ to $\mathsf{P}^{\prime\prime}\in\mathscr{P}$ is defined to be an element of the set

[TABLE]

and the composition in the category in the one given by the group structure of $\mathsf{G}({\mathbb{Q}})$. Then $\mathscr{P}$ is in fact a groupoid.

Define a functor from $\mathscr{P}$ to the category of smooth representations of $\mathsf{G}(\mathbb{A})$ as follows. It sends $\mathsf{P}^{\prime}\in\mathscr{P}$ to the representation

[TABLE]

and it sends a morphism $w:\mathsf{P}^{\prime}\rightarrow\mathsf{P}^{\prime\prime}$ to the intertwining operator

[TABLE]

This is indeed a functor by the functional equation of intertwining operators (see [La, Section 6]). Recall that the intertwining operator (18) is obtained by the meromorphic continuation of the following family of intertwining operators of convergent integrals:

[TABLE]

where $\lambda^{\prime}\in\breve{\mathfrak{a}}_{\mathsf{P}^{\prime},\mathbb{C}}$ with

[TABLE]

In (19), $\tilde{w}\in\mathsf{G}({\mathbb{Q}})$ denotes a representative of $w$, and $\operatorname{d}\!n$ denotes the quotient of the normalized Haar measures (the Haar measure on $\mathsf{N}_{\mathsf{P}^{\prime\prime}}(\mathbb{A})$ is normalized so that $\mathsf{N}_{\mathsf{P}^{\prime\prime}}({\mathbb{Q}})\backslash\mathsf{N}_{\mathsf{P}^{\prime\prime}}(\mathbb{A})$ has volume $1$, and similarly for other unipotent groups).

The theory of Eisenstein series gives a $\mathsf{G}(\mathbb{A})$-intertwining continuous linear map

[TABLE]

Recall that the map (21) is obtained by the meromorphic continuation of the following family of convergent Eisenstein series:

[TABLE]

where $\lambda^{\prime}$ is as in (20). By the functional equation of Eisenstein series, (21) for various $\mathsf{P}^{\prime}$ yields a $\mathsf{G}(\mathbb{A})$-intertwining continuous linear map

[TABLE]

The following theorem is a reformulation of a special case of [Fr, Theorem 14].

Theorem 3.2**.**

The map (22) induces a $\mathsf{G}(\mathbb{A})$-intertwining topological linear isomorphism

[TABLE]

As a reformulation of (23), we have

[TABLE]

where $\{\mathsf{P}\}\subset\mathscr{P}$ denotes the associated class of $\mathsf{P}$, namely the isomorphism class of $\mathsf{P}$ in the groupoind $\mathscr{P}$, and ${\mathrm{Aut}}_{\mathscr{P}}(\mathsf{P})$ denotes the finite group of the automorphisms of the object $\mathsf{P}$ in the groupoid $\mathscr{P}$.

3.6. Franke’s filtration

Write

[TABLE]

so that (see [Kn, Proposition 2.69] for example)

[TABLE]

Following Franke [Fr], for each real number $t\geq 0$, define

[TABLE]

[TABLE]

and

[TABLE]

where $\phi_{\mathsf{P}}\in{\mathcal{A}}_{\mathsf{P}}(\mathsf{G})$ denotes the constant term of $\phi$ along $\mathsf{P}$, as before. We remark that these spaces are independent of the pair $(\mathsf{P}_{0},\mathsf{A}_{0})$. It is clear that

[TABLE]

Define a linear map

[TABLE]

The following theorem follows from [Fr, Theorem 14] and its proof.

Theorem 3.3**.**

The linear map (25) induces an isomorphism

[TABLE]

of quasi Casselman-Wallach representations of $\mathsf{G}(\mathbb{A})$.

3.7. Franke’s filtration with fixed generalized infinitesimal character

Write

[TABLE]

for the generalized eigenspace decomposition with respect to the action of $\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})$. For each character $\chi:\operatorname{Z}(\mathfrak{g}_{\mathbb{C}})\rightarrow\mathbb{C}$,

[TABLE]

defines an increasing filtration of ${\mathcal{A}}(\mathsf{G})_{\chi}$. Let $\mathfrak{h}_{0}$ denote a maximally split Cartan subalgebra of the Lie algebra of of $\mathsf{L}_{0}(\mathbb{R})$. Then $\mathfrak{h}_{0}\supset\mathfrak{a}_{0}$ and hence $\mathfrak{h}_{0,\mathbb{C}}\supset\mathfrak{a}_{0,\mathbb{C}}$. Define

[TABLE]

This is independent of the choice of $\mathfrak{h}_{0}$. Put

[TABLE]

It is a finite set containing [math].

Lemma 3.4**.**

Suppose that $\phi\in{\mathcal{A}}(\mathsf{G})_{\chi}$ and $\phi_{\mathsf{P},\lambda}\neq 0$ for some $\mathsf{P}\in\mathscr{P}$ and $\lambda\in\breve{\mathfrak{a}}_{\mathsf{P},\mathbb{C}}$. Then

[TABLE]

Proof.

Recall that the normalized parabolic inductions for real reductive groups preserve the infinitesimal characters. This implies that

[TABLE]

Hence the lemma follows. ∎

Lemma 3.4 implies that the filtration (27) is finite. More precisely, Lemma 3.4 implies the following result.

Lemma 3.5**.**

Let $t$ be a nonnegative real number. Then

[TABLE]

4. Low degree cohomologies

We continue with the notation of the last section.

4.1. Low degree cohomologies of the space of automorphic forms

Write $\mathsf{T}_{\mathsf{P}}$ for the maximal anisotropic central torus in $\mathsf{L}_{\mathsf{P}}\cap\mathsf{G}^{\mathrm{der}}$, where $\mathsf{G}^{\mathrm{der}}$ denotes the derived subgroup of $\mathsf{G}$. Write $\mathsf{C}_{\mathsf{P}}$ for the the maximal split torus of the real algebraic torus $\mathsf{T}_{\mathsf{P}}\times_{\mathrm{spec}({\mathbb{Q}})}\mathrm{spec}(\mathbb{R})$.

We say that an irreducible Cassleman-Wallach repesentation $\sigma$ of $L_{\mathsf{P}}=\mathsf{L}_{\mathsf{P}}(\mathbb{R})$ has a $\mathfrak{g}$-real infinitesimal character if

[TABLE]

where $\mathfrak{h}$ is a Cartan subalgebra of the Lie algebra of $L_{\mathsf{P}}$, and $\lambda_{\sigma}\in\breve{\mathfrak{h}}_{\mathbb{C}}$ is a Harish-Chandra parameter of the infinitesimal character of $\sigma$. This definition is independent of $\mathfrak{h}$ and $\lambda_{\sigma}$.

Lemma 4.1**.**

Let $\sigma$ be an irreducible Cassleman-Wallach repesentation of $L_{\mathsf{P}}$ with $\mathfrak{g}$-real infinitesimal character. If it occurs as an $L_{\mathsf{P}}$-subquotient of ${\mathcal{A}}(\mathsf{L}_{\mathsf{P}})$, then the identity connected component $\mathsf{C}_{\mathsf{P}}(\mathbb{R})_{0}$ of $\mathsf{C}_{\mathsf{P}}(\mathbb{R})$ acts trivially on $\sigma$.

Proof.

The group $\mathsf{C}_{\mathsf{P}}(\mathbb{R})_{0}$ acts on $\sigma$ by a character $\chi$. The automorphic condition implies that $\chi$ is unitary, and the condition of $\mathfrak{g}$-real infinitesimal character implies that $\chi$ is real. Hence $\chi$ must be trivial. ∎

As in the Introduction, let $F$ be an irreducible finite dimensional representation of $G=\mathsf{G}(\mathbb{R})$. Recall the quantity $q_{G,F}$ from the Introduction.

Lemma 4.2**.**

Assume that $\mathsf{P}\neq\mathsf{G}$. Let $\lambda\in\breve{\mathfrak{a}}_{\mathsf{P},\mathbb{C}}$ with ${\mathrm{Re}}(\lambda)\in\overline{\breve{\mathfrak{a}}_{\mathsf{P}}^{+}}$. Let $\sigma$ be an irreducible Cassleman-Wallach representation of $L_{\mathsf{P}}$ which occurs as an irreducible $L_{\mathsf{P}}$-subquotient of ${\mathcal{A}}^{\bar{2}}(\mathsf{L}_{\mathsf{P}})_{\lambda}$. Then

[TABLE]

Proof.

If the infinitesimal character of $\sigma$ is not $\mathfrak{g}$-real, then the cohomology space of (28) vanish. So we assume that $\sigma$ has a $\mathfrak{g}$-real infinitesimal character. Then in view of the definition of $q_{G,F}$, the lemma follows from Lemma 4.1. ∎

Lemma 4.3**.**

Assume that $\mathsf{P}\neq\mathsf{G}$. Let $\lambda\in\breve{\mathfrak{a}}_{\mathsf{P},\mathbb{C}}$ with ${\mathrm{Re}}(\lambda)\in\overline{\breve{\mathfrak{a}}_{\mathsf{P}}^{+}}$. Then

[TABLE]

Proof.

Note that ${\mathcal{A}}^{\bar{2}}_{\mathsf{P}}(\mathsf{G})_{\lambda}$ has a filtration $V_{0}\subset V_{1}\subset V_{2}\subset\cdots$ such that every associated graded representation is a direct sum of representations of the form

[TABLE]

where $\sigma$ is an irreducible $\mathsf{L}_{\mathsf{P}}(\mathbb{A})$-subrepresentation of ${\mathcal{A}}^{\bar{2}}(\mathsf{L}_{\mathsf{P}})_{\lambda}$. Thus the lemma follows from Lemma 4.2. ∎

Lemma 4.4**.**

For all $i\in\mathbb{Z}$, the natural linear map

[TABLE]

is injective, and it is an isomorphism if $i<q_{G,F}$.

Proof.

In view of (24), this is implied by Lemma 4.3. ∎

Lemma 4.5**.**

The natural linear map

[TABLE]

is injective if $i\leq q_{G,F}$, and is bijective if $i<q_{G,F}$.

Proof.

This follows by Theorem 3.3, Lemma 3.5 and Lemma 4.3. ∎

Combining Lemmas 4.4 and 4.5, we get the following result.

Proposition 4.6**.**

The natural linear map

[TABLE]

is injective if $i\leq q_{G,F}$, and is bijective if $i<q_{G,F}$.

4.2. Automorphic forms on $\Gamma\backslash{{}^{\circ}\!G}$

Recall from (2) the decomposition

[TABLE]

Let $\Gamma$ be an arithmetic subgroup of $\mathsf{G}({\mathbb{Q}})$. Then $\Gamma\subset{{}^{\circ}\!G}$ and

[TABLE]

As in Section 3.1 , we say that a smooth function on $\Gamma\backslash{{}^{\circ}\!G}$ is a smooth automorphic form if it is $\operatorname{Z}({{}^{\circ}\!\mathfrak{g}_{\mathbb{C}}})$-finite, and is uniformly of moderate growth. Denote by ${\mathcal{A}}(\Gamma\backslash{{}^{\circ}\!G})$ the space of smooth automorphic forms on $\Gamma\backslash{{}^{\circ}\!G}$. As in Section 3.1, it is a quasi Casselman-Wallach representation of ${{}^{\circ}\!G}$.

Write

[TABLE]

where $\phi$ is $A_{\mathsf{G}}$-finite means that $\{\phi(\,\cdot\,a)\}_{a\in A_{\mathsf{G}}}$ spans a finite dimensional subspace of $\textit{C}^{\infty}(A_{\mathsf{G}})$. This is a quasi-Cassleman-Wallach representation of $A_{\mathsf{G}}$ under the right translations, and under the finest locally convex topology.

It is clear that

[TABLE]

Fix a ${{}^{\circ}\!G}$-invariant positive Borel measure on $\Gamma\backslash{{}^{\circ}\!G}$. It has finite total volume. Write ${\mathcal{A}}^{2}(\Gamma\backslash{{}^{\circ}\!G})\subset{\mathcal{A}}(\Gamma\backslash{{}^{\circ}\!G})$ for the subspace of square integrable functions with respect to this invariant measure, and write

[TABLE]

Lemma 4.7**.**

By restriction of continuous cohomology, the restriction map

[TABLE]

induces a linear isomorphism

[TABLE]

Likewise the restriction map

[TABLE]

induces a linear isomorphism

[TABLE]

Proof.

The group $A_{\mathsf{G}}$ acts on $F$ through a character which we denote by $\chi_{F}$. Using the Künneth formula, the lemma then follow form the fact that

[TABLE]

∎

Proposition 4.8**.**

Suppose that $\Gamma$ is a congruence subgroup of $\mathsf{G}({\mathbb{Q}})$. Then the natural linear map

[TABLE]

is injective if $i\leq q_{G,F}$, and is bijective if $i<q_{G,F}$.

Proof.

Without loss of generality, assume that $\Gamma=\mathsf{G}({\mathbb{Q}})\cap K_{f}$, where $K_{f}$ is an open compact subgroup of $\mathsf{G}(\mathbb{A}_{f})$. Then

[TABLE]

where $k$ is the cardinality of the finite set $\mathsf{G}({\mathbb{Q}})\backslash\mathsf{G}(\mathbb{A}_{f})/K_{f}$, and $\Gamma_{j}$’s are certain congruence subgroups of $\mathsf{G}({\mathbb{Q}})$ with $\Gamma_{1}=\Gamma$. Moreover,

[TABLE]

Therefore, Proposition 4.6 implies that the natural linear map

[TABLE]

is injective if $i\leq q_{G,F}$, and is bijective if $i<q_{G,F}$. The proposition then follows by using Lemma 4.7.

∎

4.3. Cohomologies of congruence groups

Let $\Gamma$ be an arithmetic subgroup of $\mathsf{G}({\mathbb{Q}})$ as before. By Shapiro’s Lemma,

[TABLE]

where the second identification is induced by the isomorphism

[TABLE]

Remark 4.9*.*

The cohomology group $\operatorname{H}^{i}(\Gamma;F)$ has a topological interpretation, as explained in what follows. Set

[TABLE]

For every open subset $U$ of $X_{\Gamma}$, write

[TABLE]

where $\tilde{U}$ denotes the pre-image of $U$ under the quotient map $G/K\rightarrow X_{\Gamma}$. Then $F_{[\Gamma]}$ is obviously a sheaf of complex vector spaces on $X_{\Gamma}$. Moreover, its sheaf cohomology agrees with the group cohomology, namely there is a vector space identification

[TABLE]

See [BW, Chapter VII] for more details.

Recall the following important result of Franke [Fr, Theorem 18] (see also [Sc, Section 13]).

Theorem 4.10**.**

Suppose that $\Gamma$ is a congruence subgroup of $\mathsf{G}({\mathbb{Q}})$. Then the embedding ${\mathcal{A}}(\Gamma\backslash{{}^{\circ}\!G})\hookrightarrow\textit{C}^{\,\infty}(\Gamma\backslash{{}^{\circ}\!G})$ induces an identification

[TABLE]

of vector spaces.

By combining (29) and Theorem 4.10, we get an identification

[TABLE]

4.4. Finite dimensional representations in ${\mathcal{A}}^{2}(\Gamma\backslash{{}^{\circ}\!G})$

In this subsection and the next one, we suppose that $\Gamma$ is a congruence subgroup of $\mathsf{G}({\mathbb{Q}})$.

Lemma 4.11**.**

There exists a cocompact closed subgroup $\Gamma^{\prime}$ of ${{}^{\circ}\!G}$ such that for every irreducible finite dimensional representation of $F^{\prime}$ of $G$, $\Gamma^{\prime}$ acts trivially on $F^{\prime\Gamma}$.

Proof.

Without loss of generality, we assume that $\mathsf{G}$ is either an algebraic torus over ${\mathbb{Q}}$, or a simply connected, connected, ${\mathbb{Q}}$-simple linear algebraic group over ${\mathbb{Q}}$. If $\mathsf{G}$ is an algebraic torus or an anisotropic simple algebraic group, then $\Gamma$ is cocompact in ${{}^{\circ}\!G}$ and there is nothing to prove. So we further assume that $\mathsf{G}$ is an isotropic simple algebraic group. Then $\Gamma$ is Zariski dense in $\mathsf{G}(\mathbb{C})$, and the representation of $G$ on $F^{\prime}$ extends to an algebraic representation of $\mathsf{G}(\mathbb{C})$. Thus the group $G$ fixes $F^{\prime\Gamma}$. ∎

As before, $\breve{F}$ denotes the dual representation of $F$.

Lemma 4.12**.**

The image of every homomorphism in ${\mathrm{Hom}}_{{}^{\circ}\!G}(\breve{F},\textit{C}^{\infty}(\Gamma\backslash{{}^{\circ}\!G}))$ is contained in ${\mathcal{A}}^{2}(\Gamma\backslash{{}^{\circ}\!G})$.

Proof.

Let $\phi\in{\mathrm{Hom}}_{{}^{\circ}\!G}(\breve{F},\textit{C}^{\infty}(\Gamma\backslash{{}^{\circ}\!G}))$. Write

[TABLE]

for the map of evaluating at $1\in\Gamma\backslash{{}^{\circ}\!G}$. Then

[TABLE]

Lemma 4.11 implies that $\mathrm{Ev}\circ\phi$ is fixed by a cocompact closed subgroup $\Gamma^{\prime}$ of ${{}^{\circ}\!G}$. Thus the image of $\phi$ is contained in the space of the bounded functions, and the lemma easily follows.

∎

By using the inner product induced by the invariant measure, we know that the representation ${\mathcal{A}}^{2}(\Gamma\backslash{{}^{\circ}\!G})$ of ${{}^{\circ}\!G}$ is completely reducible. By Lemma 4.12, we have that

[TABLE]

4.5. Proof of Theorem 1.7

Recall the linear map (4) of the Introduction:

[TABLE]

Recall the identification (31). One checks that the map (32) is identical to the map

[TABLE]

induced by the linear map

[TABLE]

Since the representation ${\mathcal{A}}^{2}(\Gamma\backslash{{}^{\circ}\!G})$ of ${{}^{\circ}\!G}$ is completely reducible, the natural linear map

[TABLE]

is always injective, and it is bijective if $i<r_{{{}^{\circ}\!G},F}=r_{G,F}$. Together with Proposition 4.8, this proves Theorem 1.7.

5. A result on semisimple real Lie algebras

The notation of this section is independent of the previous ones. Let $\mathfrak{g}$ be a semisimple real Lie algebra.

5.1. A quantity $r_{\mathfrak{g}}$

Fix a Cartan involution

[TABLE]

and write

[TABLE]

for the corresponding Cartan decomposition. As before, we use a subscript $\mathbb{C}$ to indicate the complexification of a real vector space. Then we have the complexifications

[TABLE]

and

[TABLE]

If $\mathfrak{g}$ is compact, namely it is isomorphic to the Lie algebra of a compact Lie group, we set $r_{\mathfrak{g}}:=+\infty$. If $\mathfrak{g}$ is noncompact and simple, set

[TABLE]

where $\mathfrak{n}_{\mathfrak{q}}$ denotes the nilpotent radical of $\mathfrak{q}$. In general when $\mathfrak{g}$ is noncompact, define

[TABLE]

In view of the unitarizability Theorem of Vogan [Vo] and Wallach [Wa3], the following result is a consequence of Vogan-Zuckerman’s theory of unitary representations with nonzero cohomology (see [Ku] and [VZ, Theorem 8.1]).

Proposition 5.1**.**

Let $G$ be a real reductive group and define $r_{G}$ as in the Introduction. Then

[TABLE]

5.2. A quantity $r_{\mathfrak{g},\mu}$

Let $\mathfrak{t}$ be a Cartan subalgebra of $\mathfrak{k}$. Write $\mathfrak{a}_{0}$ for the centralizer of $\mathfrak{t}$ in $\mathfrak{s}$ so that $\mathfrak{h}_{0}:=\mathfrak{t}\oplus\mathfrak{a}_{0}$ is a fundamental Cartan subalgebra of $\mathfrak{g}$. Write $\Delta(\mathfrak{t}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ for the root system of $\mathfrak{g}_{\mathbb{C}}$ with respect to $\mathfrak{t}_{\mathbb{C}}$, and likewise write $\Delta(\mathfrak{h}_{0,\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ for the root system of $\mathfrak{g}_{\mathbb{C}}$ with respect to $\mathfrak{h}_{0,\mathbb{C}}$.

Definition 5.2**.**

An irreducible finite dimensional representation of $\mathfrak{g}$ is said to be pure if its infinitesimal character has a Harish-Chandra parameter in $\breve{\mathfrak{t}}_{\mathbb{C}}\subset\breve{\mathfrak{h}}_{0,\mathbb{C}}$.

Lemma 5.3**.**

Let $F$ be an irreducible finite dimensional representation of $\mathfrak{g}$. Then $F$ is pure if and only if there exists a pair $(G,\pi)$ such that

• •

$G$* is a connected semsimple Lie group with finite center whose Lie algebra is identified with $\mathfrak{g}$;*

• •

the representation $F$ of $\mathfrak{g}$ integrates to a representation of $G$;

• •

$\pi$* is an irreducible unitarizable Casselman-Wallach representation of $G$ such that the total cohomology group is nonzero:*

[TABLE]

Proof.

This is implied by [VZ, Theorem 5.6]. ∎

Let $\mu\in\breve{\mathfrak{t}}_{\mathbb{C}}\subset\breve{\mathfrak{h}}_{0,\mathbb{C}}$. Suppose that it is regular integral (with respect to $\mathfrak{g}_{\mathbb{C}}$) in the sense that

[TABLE]

Here and as before, $\alpha^{\vee}\in\mathfrak{h}_{0,\mathbb{C}}$ denotes the coroot corresponding to the root $\alpha$.

The functional $\mu$ determines a positive system

[TABLE]

Denote by $\rho_{\mu}\in\breve{\mathfrak{t}}_{\mathbb{C}}$ the half sum of the roots in $\Delta(\mathfrak{t}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})^{\mu,+}$ (counted with multiplicities). Then $\mu-\rho_{\mu}$ determines a $\theta$-stable nilpotent subalgebra $\mathfrak{n}(\mu)$ of $\mathfrak{g}_{\mathbb{C}}$ whose weights are

[TABLE]

Set

[TABLE]

Lemma 5.4**.**

Let $G$ be a connected semsimple Lie group with finite center whose Lie algebra is identified with $\mathfrak{g}$. Let $F$ be an irreducible finite dimensional representation of $G$, and let $\pi$ be an irreducible unitarizable Casselman-Wallach representation of $G$. If

[TABLE]

then there exists $\lambda\in\breve{\mathfrak{t}}_{\mathbb{C}}\subset\breve{\mathfrak{h}}_{0,\mathbb{C}}$ which is a Harish-Chandra parameter of the infinitesimal character of $\breve{F}$ such that

[TABLE]

Proof.

This is implied by [VZ, Theorem 5.6] and [Wa1, Theorem 9.6.6]. ∎

5.3. The result

Suppose that $\mathfrak{g}$ is noncompact so that it has a proper parabolic subalgebra $\mathfrak{p}=\mathfrak{l}\ltimes\mathfrak{n}$, where $\mathfrak{n}$ is the nilpotent radical of $\mathfrak{p}$, and $\mathfrak{l}=\mathfrak{p}\cap\theta(\mathfrak{p})$. Write

[TABLE]

where $\mathfrak{m}:=[\mathfrak{l},\mathfrak{l}]$ is semisimple, $\mathfrak{z}$ is the center of $\mathfrak{l}$, $\mathfrak{c}:=\mathfrak{z}\cap\mathfrak{k}$ and $\mathfrak{a}:=\mathfrak{z}\cap\mathfrak{s}$.

Let $\mathfrak{c}^{\prime}$ be a Cartan subalgebra of $\mathfrak{k}\cap\mathfrak{m}$, and write $\mathfrak{a}^{\prime}$ for its centralizer in $\mathfrak{s}\cap\mathfrak{m}$. Then $\mathfrak{c}^{\prime}\oplus\mathfrak{a}^{\prime}$ is a fundamental Cartan subalgebra of $\mathfrak{m}$, and

[TABLE]

is a Cartan subalgebra of $\mathfrak{g}$. We have the complexifications

[TABLE]

Let $\lambda$ be a linear functional on $\mathfrak{h}_{\mathbb{C}}$ which is regular integral in the sense that

[TABLE]

Then $\lambda|_{\mathfrak{a}_{\mathbb{C}}}\in\breve{\mathfrak{a}}_{\mathbb{C}}\subset\breve{\mathfrak{h}}_{\mathbb{C}}$, and

[TABLE]

for all $\alpha\in\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{n}_{\mathbb{C}})$. Here and as usual, $\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{n}_{\mathbb{C}})\subset\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ denotes the roots associated to $\mathfrak{n}_{\mathbb{C}}$. The rest of this section is devoted to a proof of the following estimate.

Proposition 5.5**.**

Let the notation and the assumptions be as above. Assume that

• •

$\langle\lambda|_{\mathfrak{a}_{\mathbb{C}}},\alpha^{\vee}\rangle\geq 0$, for all $\alpha\in\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{n}_{\mathbb{C}})$, and

• •

$\lambda|_{\mathfrak{a}^{\prime}_{\mathbb{C}}}=0$.

Then

[TABLE]

Note that (38) and the second assumption of Proposition 5.5 imply that $\lambda|_{\mathfrak{c}^{\prime}_{\mathbb{C}}}$ is regular integral with respcet to $\mathfrak{m}_{\mathbb{C}}$. Thus the quantity $r_{\mathfrak{m},\lambda|_{\mathfrak{c}^{\prime}_{\mathbb{C}}}}$ is defined as in (35).

5.4. Another quantity $r^{\prime}_{\mathfrak{g}}$

As before we assume that the semisimple Lie algebra $\mathfrak{g}$ is noncompact. Set

[TABLE]

where $\mathfrak{n}_{\mathfrak{q}}$ denotes the nilpotent radical of $\mathfrak{q}$. Using the explicit data of noncompact simple real Lie algebras given in [Kn, Appendix C, Sections 3 and 4], it is routine to calculate $r^{\prime}_{\mathfrak{g}}$ for all noncompact simple real Lie algebras. On the other hand, the quantity $r_{\mathfrak{g}}$ is calculated in [En, Table 1], [Ku, Theorem 2, Theorem 3], [VZ, Table 8.2] and [LS2, Section 2.B]. We summarize the results in the following Tables 1, 2 and 3. Cartan labels are used to denote the noncompact noncomplex exceptional simple real Lie algebras, as in [Kn, Appendix C, Section 4].

We observe that in all cases, $r_{\mathfrak{g}}^{\prime}\geq r_{\mathfrak{g}}$.

5.5. A proof of Proposition 5.5

We continue with the notation and the assumptions of Proposition 5.5. For the proof of Proposition 5.5, we assume without loss of generality that $\mathfrak{g}$ is simple.

Lemma 5.6**.**

One has that

[TABLE]

Proof.

Note that the parabolic subalgebra $\theta(\mathfrak{p})$ is opposite to $\mathfrak{p}$. Thus

[TABLE]

and consequently

[TABLE]

where $\theta:\breve{\mathfrak{h}}_{\mathbb{C}}\rightarrow\breve{\mathfrak{h}}_{\mathbb{C}}$ is the linear map induced by $\theta:\mathfrak{g}_{\mathbb{C}}\rightarrow\mathfrak{g}_{\mathbb{C}}$. This implies that

[TABLE]

and consequently,

[TABLE]

Recall that $\lambda$ is regular integral, and note that

[TABLE]

Therefore we have that

[TABLE]

This proves the lemma.

∎

Lemma 5.7**.**

If $\mathfrak{g}$ is not isomorphic to a split simple real Lie algebra of type $A_{n}$ ($n\geq 2$), $D_{n}$ ($n\geq 4$), $E_{6}$ or $E_{7}$, then

[TABLE]

Proof.

This follows form Tables 1, 2 and 3. ∎

By Lemma 5.6 and Lemma 5.7, Proposition 5.5 holds when $\mathfrak{g}$ is not isomorphic to a split simple real Lie algebra of type $A_{n}$ ($n\geq 2$), $D_{n}$ ($n\geq 4$), $E_{6}$ or $E_{7}$.

We will need the following general result.

Lemma 5.8**.**

Let $\mathfrak{l}^{\prime}_{\mathbb{C}}$ be a reductive finite dimensional complex Lie algebra, and let $F^{\prime}$ be an irreducible finite dimensional representation of $\mathfrak{l}^{\prime}_{\mathbb{C}}$. Let $\mathfrak{h}^{\prime}_{\mathbb{C}}$ be a Cartan subalgebra of $\mathfrak{l}^{\prime}_{\mathbb{C}}$ and fix a positive system $\Delta(\mathfrak{h}^{\prime}_{\mathbb{C}},\mathfrak{l}^{\prime}_{\mathbb{C}})^{+}$ of the root system $\Delta(\mathfrak{h}^{\prime}_{\mathbb{C}},\mathfrak{l}^{\prime}_{\mathbb{C}})$. Write $\rho^{\prime\vee}\in\mathfrak{h}^{\prime}_{\mathbb{C}}$ for the half sum of the positive coroots associated to $\Delta(\mathfrak{h}^{\prime}_{\mathbb{C}},\mathfrak{l}^{\prime}_{\mathbb{C}})^{+}$. Then the set of eigenvalues of $\rho^{\prime\vee}$ on $F^{\prime}$ has the form

[TABLE]

where $b_{0}\in\frac{1}{2}\mathbb{Z}$, and $c_{0}\geq 0$ is an integers.

Proof.

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{k}$ ($k\geq 0$) be the simple roots in $\Delta(\mathfrak{h}^{\prime}_{\mathbb{C}},\mathfrak{l}^{\prime}_{\mathbb{C}})^{+}$. Let $X_{j}$ be a root vector attached to $\alpha_{j}$ ($1\leq j\leq k$). Let $v_{0}$ be a lowest weight vector in $F^{\prime}$. Then

[TABLE]

for some $b_{0}\in\frac{1}{2}\mathbb{Z}$. Note that $F^{\prime}$ is spanned by vectors of the form

[TABLE]

where $r\geq 0$, $1\leq j_{1},j_{2},\ldots,j_{r}\leq k$. The lemma then easily follows by noting that

[TABLE]

∎

Let $\check{\mathfrak{g}}_{\mathbb{C}}$ denote the Langlands dual Lie algebra of $\mathfrak{g}_{\mathbb{C}}$, with Cartan subalgebra $\check{\mathfrak{h}}_{\mathbb{C}}$ equals the dual of $\mathfrak{h}_{\mathbb{C}}$. Let $\check{\mathfrak{p}}_{\mathbb{C}}=\check{\mathfrak{l}}_{\mathbb{C}}\ltimes\check{\mathfrak{n}}_{\mathbb{C}}$ denote the parabolic subalgebras of $\check{\mathfrak{g}}_{\mathbb{C}}$ corresponding to $\mathfrak{p}_{\mathbb{C}}$, where $\check{\mathfrak{l}}_{\mathbb{C}}\supset\check{\mathfrak{h}}_{\mathbb{C}}$ is the Levi factor corresponding to $\mathfrak{l}_{\mathbb{C}}$, and $\check{\mathfrak{n}}_{\mathbb{C}}$ is the nilpotent radical corresponding to $\mathfrak{n}_{\mathbb{C}}$. Then $\check{\mathfrak{m}}_{\mathbb{C}}:=[\check{\mathfrak{l}}_{\mathbb{C}},\check{\mathfrak{l}}_{\mathbb{C}}]$ is the Langlands dual of $\mathfrak{m}_{\mathbb{C}}$.

Lemma 5.9**.**

Assume that $\mathfrak{m}$ is nonzero and contains no compact simple ideal, and that the adjoint representation of $\check{\mathfrak{l}}_{\mathbb{C}}$ on $\check{\mathfrak{n}}_{\mathbb{C}}$ is irreducible. Then

[TABLE]

or

[TABLE]

Proof.

For simplicity write $\lambda^{\prime}:=\lambda|_{\mathfrak{c}^{\prime}_{\mathbb{C}}}$. It is regular integral with respect to $\mathfrak{m}_{\mathbb{C}}$. As in (34) we define the positive root system $\Delta(\mathfrak{c}^{\prime}_{\mathbb{C}},\mathfrak{m}_{\mathbb{C}})^{\lambda^{\prime},+}$, and write $\rho_{\lambda^{\prime}}\in\breve{\mathfrak{c}^{\prime}}_{\mathbb{C}}$ for the half sum of the positive roots (counted with multiplicities).

If $\lambda^{\prime}\neq\rho_{\lambda^{\prime}}$, then the assumption on $\mathfrak{m}$ implies that

[TABLE]

So we assume that $\lambda^{\prime}=\rho_{\lambda^{\prime}}$.

Write

[TABLE]

Note that $\check{\mathfrak{h}}_{\mathbb{C}}=\breve{\mathfrak{h}}_{\mathbb{C}}$, and $\breve{\mathfrak{c}}_{\mathbb{C}}\oplus\breve{\mathfrak{a}}_{\mathbb{C}}$ is identified with the center of $\check{\mathfrak{l}}_{\mathbb{C}}$. By Schur’s lemma, the irreducibility assumption implies that $\lambda|_{\mathfrak{c}_{\mathbb{C}}\oplus\mathfrak{a}_{\mathbb{C}}}\in\breve{\mathfrak{c}}_{\mathbb{C}}\oplus\breve{\mathfrak{a}}_{\mathbb{C}}\subset\check{\mathfrak{l}}_{\mathbb{C}}$ acts on $\check{\mathfrak{n}}_{\mathbb{C}}$ by the multiplication of a constant $a_{0}\in\mathbb{C}$. Thus

[TABLE]

Note that

[TABLE]

is also the half sum of a positive system of the root system $\Delta(\mathfrak{h}_{\mathbb{C}},\mathfrak{l}_{\mathbb{C}})$. We identify it with an element of $\check{\mathfrak{h}}_{\mathbb{C}}$ which is the half sum of the coroots associated to a positive system of the root system $\Delta(\check{\mathfrak{h}}_{\mathbb{C}},\check{\mathfrak{l}}_{\mathbb{C}})$. Then by Lemma 5.8,

[TABLE]

where $b_{0}\in\frac{1}{2}\mathbb{Z}$, and $c_{0}\geq 0$ is an integers.

Since $\lambda|_{\mathfrak{c}^{\prime}_{\mathbb{C}}}=\rho_{\lambda^{\prime}}$ and $\lambda|_{\mathfrak{a}^{\prime}_{\mathbb{C}}}=0$, by combining (39) and (40), we get that

[TABLE]

Because $\lambda$ is regular integral, the above set either consists of only positive integers, or consists only negative integers. By Lemma 5.6, it has to consists only of positive integers. This proves the lemma. ∎

Lemma 5.10**.**

(a). Assume that $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{n}(\mathbb{R})$ ($n\geq 3$). If $\mathfrak{p}$ is not a maximal parabolic subalgebra, then

[TABLE]

If $\mathfrak{p}$ is a maximal parabolic subalgebra, then

[TABLE]

(b) Assume that $\mathfrak{g}=\mathfrak{s}\mathfrak{o}(n,n)$ ($n\geq 4$). If $\mathfrak{m}$ is not isomorphic to $\mathfrak{s}\mathfrak{o}(n-1,n-1)$ or $\mathfrak{s}\mathfrak{l}_{n}(\mathbb{R})$, then

[TABLE]

If $\mathfrak{m}$ is isomorphic to $\mathfrak{s}\mathfrak{o}(n-1,n-1)$ or $\mathfrak{s}\mathfrak{l}_{n-1}(\mathbb{R})$, then

[TABLE]

(c) Assume that $\mathfrak{g}$ is split of type $E_{6}$. If $\mathfrak{m}$ is not isomorphic to $\mathfrak{s}\mathfrak{o}(5,5)$, then

[TABLE]

If $\mathfrak{m}$ is isomorphic to $\mathfrak{s}\mathfrak{o}(5,5)$, then

[TABLE]

(d) Assume that $\mathfrak{g}$ is split of type $E_{7}$. If $\mathfrak{m}$ is not isomorphic to the split simple Lie algebra of type $E_{6}$, then

[TABLE]

If $\mathfrak{m}$ is isomorphic to the split simple Lie algebra of type $E_{6}$, then

[TABLE]

Proof.

This is routine to check. ∎

Lemma 5.11**.**

Proposition 5.5 holds when $\mathfrak{g}$ is isomorphic to $\mathfrak{s}\mathfrak{l}_{n}(\mathbb{R})$ ($n\geq 3$).

Proof.

If $\mathfrak{p}$ is not a maximal parabolic subalgebra, then the lemma follows form Lemma 5.6 and the first assertion of part (a) of Lemma 5.10. If $\mathfrak{p}$ is a maximal parabolic subalgebra, then the representation of $\check{\mathfrak{l}}_{\mathbb{C}}$ on $\check{\mathfrak{n}}_{\mathbb{C}}$ is irreducible. The lemma then follows from Lemma 5.6, Lemma 5.9, and the second assertion of part (a) of Lemma 5.10.

∎

The same proof as Lemma 5.11 shows that Proposition 5.5 also holds when $\mathfrak{g}$ is isomorphic to a split simple real Lie algebra of type $D_{n}$ ($n\geq 4$), $E_{6}$ or $E_{7}$. This completes the proof of Proposition 5.5.

6. A proof of Theorem 1.4

Let the notation be as in Section 1. To prove Theorem 1.4, we assume without loss of generality that $G$ is connected and semisimple. As in Theorem 1.4, $P$ is a proper parabolic subgroup of $G$, $N$ is its unipotent radical, and $L:=P/N$.

Shapiro’s lemma implies that

[TABLE]

where $\rho_{P}$ is the positive character of $P$ whose square equals the modular character of $P$. We have a spectral sequence

[TABLE]

Write $L_{0}$ for the identity connected component of $L$. Note that $\operatorname{H}^{j}(\mathfrak{n}_{\mathbb{C}};F)$ is completely reducible as a representation of $L_{0}$. Assume that

[TABLE]

for some $i,j\in\mathbb{Z}$. Then

[TABLE]

for some unitarizable irreducible Casselman-Wallach representations $\sigma_{0}$ of $L_{0}$, and some irreducible $L_{0}$-subrepresentation $F_{0}$ of $\operatorname{H}^{j}(\mathfrak{n}_{\mathbb{C}};F)$.

Fix a Cartan involution $\theta:G\rightarrow G$, and still denote its differential by $\theta:\mathfrak{g}\rightarrow\mathfrak{g}$. Identify $L$ with $P\cap\theta(P)$. Then its Lie algebra $\mathfrak{l}$ has a decomposition

[TABLE]

as in (36). Let

[TABLE]

be a Cartan subalgebra of $\mathfrak{l}$ which is obtained as in (37).

By Lemma 5.4, there exists $\lambda\in\breve{\mathfrak{h}}_{\mathbb{C}}$ such that

• •

$-\lambda$ is a Harish-Chandra parameter of the infinitesimal character of the representation $F_{0}\otimes\rho_{P}$ of $L_{0}$;

• •

$\lambda|_{\mathfrak{a}^{\prime}_{\mathbb{C}}}=0$; and

• •

$i\geq r_{\mathfrak{m},\lambda|_{\mathfrak{c}^{\prime}_{\mathbb{C}}}}$.

The first of the above three conditions in particular implies that

[TABLE]

By Casselman-Osborne Theorem (see [KV, Theorem 4.149]), $-\lambda$ is also a Harish-Chandra parameter of the infinitesimal character of the representation $F$ of $G$. Hence $\lambda$ is regular integral.

Lemma 6.1**.**

The assumption that $F_{0}$ is an irreducible $L_{0}$-subrepresentation of $\operatorname{H}^{j}(\mathfrak{n}_{\mathbb{C}};F)$ implies that

[TABLE]

Proof.

This follows from Kostant’s Theorem (see [KV, Theorem 4.139]). ∎

Now Proposition 5.5 implies that

[TABLE]

This proves Theorem 1.4.

Acknowledgements

Jian-Shu Li’s research was partially supported by RGC-GRF grant 16303314 of HKSAR.

Binyong Sun was supported in part by the National Natural Science Foundation of China (No. 11525105, 11688101, 11621061 and 11531008).