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

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.