Smooth cuspidal automorphic forms and integrable discrete series

TL;DR

This paper constructs smooth cuspidal automorphic forms associated with integrable discrete series for semisimple Lie groups, utilizing Schwartz space theory to bridge classical and adelic contexts.

Contribution

It introduces a method to explicitly construct smooth cuspidal automorphic forms from integrable discrete series using Schwartz space theory, extending previous frameworks.

Findings

Explicit continuous map from smooth vectors to Schwartz space duals

Construction applicable to classical and adelic cases

Bridges integrable discrete series with automorphic forms

Abstract

In this paper we construct smooth cuspidal automorphic forms related to integrable discrete series of a connected semisimple Lie group with finite center for classical and adelic situation as an application of the theory of Schwartz spaces for automorphic forms developed by Casselman. In the classical situation, smooth cuspidal automorphic forms are constructed via an explicit continuous map from the Frech\' et space of smooth vectors of a Banach realization inside $L^1(G)$ of an integrable discrete series into the space of smooth vectors of a strong topological dual of an appropriate Schwartz space.