On the Schwartz space ${\mathcal {S}}(G(k)\backslash G({\mathbb {A}}))$

Abstract

For a connected reductive group G defined over a number field k, we construct the Schwartz space ${\mathcal {S}}(G(k)\backslash G({\mathbb {A}}))$. This space is an adelic version of Casselman’s Schwartz space ${\mathcal {S}}({\Gamma }\backslash G_\infty )$, where ${\Gamma }$ is a discrete subgroup of $G_\infty :=\prod _{v\in V_\infty }G(k_v)$. We study the space of tempered distributions ${\mathcal {S}}(G(k)\backslash G({\mathbb {A}}))'$ and investigate applications to automorphic forms on $G({\mathbb {A}})$. In particular, we study the representation $\left( r',{\mathcal {S}}(G(k)\backslash G({\mathbb {A}}))'\right)$ contragredient to the right regular representation $(r,{\mathcal {S}}(G(k)\backslash G({\mathbb {A}})))$ of $G({\mathbb {A}})$ and describe the closed irreducible admissible subrepresentations of ${\mathcal {S}}(G(k)\backslash G({\mathbb {A}}))'$ assuming that G is semisimple.

Appendix

Here we collect a few facts from functional analysis used in the paper. All vector spaces are assumed to be complex. All locally convex topological vector spaces are assumed to be Hausdorff.

We start by recalling the definition and basic properties of the Gelfand–Pettis integral (e.g., see [13, §14]).

Theorem 4

Let X be a compact Hausdorff space with a Radon measure dx, and let E and F be quasi-complete (e.g., complete) locally convex topological vector spaces. Then, for every continuous function $f:X\rightarrow E$ there exists a unique $\int _Xf(x)\,dx\in E$ (the Gelfand–Pettis integral of f) such that for every continuous linear functional $T:E\rightarrow {\mathbb {C}}$, we have

$$\begin{aligned} \left<T,\int _Xf(x)\,dx\right>=\int _X\left<T,f(x)\right>\,dx. \end{aligned}$$

(70)

Moreover, the following holds:

• (1) For every continuous linear operator $A:E\rightarrow F$,

  $$\begin{aligned} A\left( \int _Xf(x)\,dx\right) =\int _XA(f(x))\,dx. \end{aligned}$$

  (71)

• (2) For every continuous seminorm $\nu :E\rightarrow {\mathbb {R}}_{\ge 0}$,

  $$\begin{aligned} \left\Vert \int _Xf(x)\,dx\right\Vert _\nu \le \int _X\left\Vert f(x)\right\Vert _\nu \,dx. \end{aligned}$$

Next, we recall the definition and basic properties of LF-spaces—strict inductive limits of increasing sequences of Fréchet spaces (see e.g. [36, §13], [33, §II.6.3] or [30, §12.1]).

Definition 3

Let $(E_m)_{m\in {\mathbb {Z}}_{>0}}$ be a sequence of Fréchet spaces such that $E_m$ is a closed subspace of $E_{m+1}$ for every $m\in {\mathbb {Z}}_{>0}$. The vector space $E:=\bigcup _{m\in {\mathbb {Z}}_{>0}}E_m$ equipped with the finest locally convex topology with respect to which the inclusion maps $E_m\hookrightarrow E$ are continuous is called the LF-space with a defining sequence $(E_m)_{m\in {\mathbb {Z}}_{>0}}$.

The next lemma uses the notion of bounded sets in a locally convex topological vector space. We recall their definition [30, Definition 6.1.1 and Theorem 6.1.5].

Definition 4

Let E be a locally convex topological vector space, and let ${\mathcal {C}}$ be a family of continuous seminorms generating its topology. A subset $B\subseteq E$ is bounded in E if the following equivalent conditions hold:

• (1) For every neighborhood U of 0 in E, there exists $t_0\in {\mathbb {R}}_{>0}$ such that $B\subseteq tU$ for all $t\in {\mathbb {C}}$ such that $\left|t\right|\ge t_0$.

• (2) $\sup _{v\in B}\left\Vert v\right\Vert _\rho <\infty$ for all $\rho \in {\mathcal {C}}$.

We also need the following definition.

Definition 5

Let E be a vector space. The absolutely convex hull of a subset $A\subseteq E$ is the set

$$\begin{aligned} {\text {AConv}}(A):=\left\{ \sum _{i=1}^na_iv_i:n\in {\mathbb {Z}}_{>0},\ v_i\in A,\ a_i\in {\mathbb {C}},\ \sum _{i=1}^n\left|a_i\right|\le 1\right\} . \end{aligned}$$

Lemma 24

Let E be an LF-space with a defining sequence $(E_m)_{m\in {\mathbb {Z}}_{>0}}$. Then, we have the following:

• (1) The space E is a complete locally convex topological vector space.

• (2) For every $m\in {\mathbb {Z}}_{>0}$, let ${\mathcal {U}}_m$ be a neighborhood basis of 0 in $E_m$. Let ${\mathcal {U}}$ be the family of subsets $U\subseteq E$ of the form

  $$\begin{aligned} U={\text {AConv}}\left( \bigcup _{m\in {\mathbb {Z}}_{>0}}U_m\right) , \end{aligned}$$

  where $U_m\in {\mathcal {U}}_m$. Then, ${\mathcal {U}}$ is a neighborhood basis of 0 in E.

• (3) For every $m\in {\mathbb {Z}}_{>0}$, $E_m$ is a closed subspace of E.

• (4) A subset B of E is bounded in E if and only if there exists $m\in {\mathbb {Z}}_{>0}$ such that $B\subseteq E_m$ and B is bounded in $E_m$.

• (5) Let $(v_k)_{k\in {\mathbb {Z}}_{>0}}\subseteq E$ and $v\in E$. Then, $v_k\rightarrow v$ in E if and only if there exists $m\in {\mathbb {Z}}_{>0}$ such that $(v_k)_{k\in {\mathbb {Z}}_{>0}}\subseteq E_m$, $v\in E_m$, and $v_k\rightarrow v$ in $E_m$.

• (6) Let F be a locally convex topological vector space. Then, a linear operator $A:E\rightarrow F$ is continuous if and only if the restrictions $A\big |_{E_m}:E_m\rightarrow F$, $m\in {\mathbb {Z}}_{>0}$, are continuous.

• (7) A seminorm $p:E\rightarrow {\mathbb {R}}_{\ge 0}$ is continuous if and only if the restrictions $p\big |_{E_m}:E_m\rightarrow {\mathbb {R}}_{\ge 0}$, $m\in {\mathbb {Z}}_{>0}$, are continuous.

Proof

The claim (1) holds because E is locally convex by definition (it is Hausdorff by [30, Theorem 12.1.3(b)]) and complete by [30, Theorem 12.1.10]; (2) follows from [30, Theorems 12.1.1 and 4.2.11]; (3) holds by [30, Theorem 12.1.3(a)], (4) by [30, Theorem 12.1.7(a)]; (5) by [30, Theorem 12.1.7(b)], (6) by [30, Theorem 12.2.2], and (7) follows easily from the definition of topology on E. $\square$

Next, we recall the notion of the strong dual of a locally convex topological vector space [33, §IV.5] (see also [32, Theorem 3.18]).

Definition 6

Let E be a locally convex topological vector space. The strong dual $E'$ of E is the space of continuous linear functionals $E\rightarrow {\mathbb {C}}$ equipped with the locally convex topology generated by the seminorms $\left\Vert {\,\cdot \,}\right\Vert _B:E'\rightarrow {\mathbb {R}}_{\ge 0}$,

$$\begin{aligned} \left\Vert T\right\Vert _B:=\sup _{f\in B}\left|\left<T,f\right>\right| \end{aligned}$$

(72)

where B goes over all bounded sets in E.

Lemma 25

[36, Corollary 2 of Theorem 32.2] Let E be a metrizable locally convex topological vector space (e.g., a Fréchet space) or an LF-space. Then, the strong dual $E'$ is complete.

Lemma 26

([19, §29.1(7)]) Let E be a metrizable locally convex topological vector space. Then, the strong dual $E'$ is metrizable if and only if E is normable.

Lemma 27

Let $E_1,\ldots ,E_n$ be locally convex topological vector spaces and for each $i\in \left\{ 1,\ldots ,n\right\}$, let $\iota _i$ be the canonical inclusion $E_i\rightarrow \bigoplus _{i=1}^n E_i$. Let us equip the direct sums $\bigoplus _{i=1}^nE_i$ and $\bigoplus _{i=1}^nE_i'$ with product topologies. Then, we have the following:

• (1) A subset B of $\bigoplus _{i=1}^nE_i$ is bounded if and only if $B\subseteq \bigoplus _{i=1}^n B_i$ for some bounded subsets $B_i$ of $E_i$.

• (2) The linear operator

  $$\begin{aligned} {\Psi }:\left( \bigoplus _{i=1}^nE_i\right) '\rightarrow \bigoplus _{i=1}^nE_i',\qquad T\mapsto \left( T\circ \iota _i\right) _{i=1}^n, \end{aligned}$$

  is an isomorphism of topological vector spaces.

Proof

(1) This is a special case of [33, I.5.5].

(2) It is elementary and well-known that ${\Psi }$ is a linear isomorphism. It follows easily from (1) that it is also a homeomorphism. $\square$