Let \(G\) be a connected reductive group defined over a number field \(F\) with adele ring \(\mathbb A\). We will introduce the Schwartz space \(S(G(F)\backslash G(\mathbb A))\) -- an adelic version of Casselman's Schwartz space \(S(H\backslash G_\infty)\), where \(H\) is a discrete subgroup of \(G_\infty:=\prod_{v|\infty}G(F_v)\). The strong dual \(S(G(F)\backslash G(\mathbb A))'\) has many intriguing properties, e.g., its (naturally defined) Gårding subspace may be identified with the space \(A_{umg}(G(F)\backslash G(\mathbb A))\) of smooth functions of uniform moderate growth. We will describe the closed irreducible admissible \(G(\mathbb A)\)-invariant subspaces of \(S(G(F)\backslash G(\mathbb A))'\) and discuss applications to automorphic forms. This is joint work with Goran Muic.
On the Schwartz space \(S(G(F)\backslash G(\mathbb A))\)
03.03.2020 13:15 - 14:45
Organiser:
H. Grobner, A. Minguez-Espallargas, A. Mellit
Location: