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For a connected reductive group $ G $ defined over a number field $ k $, weconstruct 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 discretesubgroup of $ G_\infty:=\prod_{v\in V_\infty}G(k_v) $. We study the space oftempered distributions $ \mathcal{S}(G(k)\backslash G(\mathbb A))' $ andinvestigate applications to automorphic forms on $ G(\mathbb A) $. Inparticular, we study the representation $ \left(r',\mathcal{S}(G(k)\backslashG(\mathbb{A}))'\right) $ contragredient to the right regular representation $(r,\mathcal{S}(G(k)\backslash G(\mathbb{A}))) $ of $ G(\mathbb{A}) $ anddescribe the closed irreducible admissible subrepresentations of $\mathcal{S}(G(k)\backslash G(\mathbb{A}))' $ assuming that $ G $ is semisimple.

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