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

## TL;DR

This paper constructs and analyzes a Schwartz space for automorphic forms on reductive groups over number fields, extending classical spaces and exploring their distributional and representation-theoretic properties.

## Contribution

It introduces an adelic Schwartz space for reductive groups over number fields and studies its distributional dual and automorphic representations, extending Casselman's classical framework.

## Key findings

- Constructed the adelic Schwartz space for reductive groups.
- Analyzed the space of tempered distributions and their automorphic applications.
- Described irreducible admissible subrepresentations of the dual space.

## 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.