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 \( {\mathcal {S}}(G(k)\backslash G({\mathbb {A}}))' \) 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.