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.

中文翻译：

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在施瓦茨空间 $$ {\mathcal {S}}(G(k)\backslash G({\mathbb {A}})) $$S(G(k)\G(A))

对于定义在数域 $ k $ 上的连通还原群 $ G $，我们构造 Schwartz 空间 $ \mathcal{S}(G(k)\backslash G(\mathbb{A})) $。该空间是 Casselman 的 Schwartz 空间 $\mathcal{S}(\Gamma\backslash G_\infty) $ 的 adelic 版本，其中 $ \Gamma $ 是 $ G_\infty:=\prod_{v\in V_ 的离散子群\infty}G(k_v) $。我们研究了调和分布 $\mathcal{S}(G(k)\backslash G(\mathbb A))' $ 的空间，并研究了 $ G(\mathbb A) $ 上自守形式的应用。特别地，我们研究了表示 $ \left(r',\mathcal{S}(G(k)\backslash G(\mathbb{A}))'\right) $ 与正确的正则表示 $ (r,