  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.
