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 $\mathcal{S}(G(k)\backslash G(\mathbb{A}){)}^{\prime }$ and investigate applications to automorphic forms on $G(\mathbb{A})$. In particular, we study the representation $\left(r^{\prime },\mathcal{S}(G(k)\backslash G(\mathbb{A}){)}^{\prime }\right)$ 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 $\mathcal{S}(G(k)\backslash G(\mathbb{A}){)}^{\prime }$ assuming that $G$ is semisimple.

41 pages