On interactions between harmonic analysis and the theory of automorphic forms

TL;DR

This paper reviews the interplay between harmonic analysis and automorphic forms, illustrating how each field informs the other through examples involving classical groups and their duals.

Contribution

It highlights the connections between harmonic analysis and automorphic forms, emphasizing the role of specific representations like isolated points in duals.

Findings

Harmonic analysis aids in understanding automorphic representations.

Automorphic forms influence harmonic analysis techniques.

Key representations are isolated points in duals.

Abstract

In this paper we review some connections between harmonic analysis and the modern theory of automorphic forms. We indicate in some examples how the study of problems of harmonic analysis brings us to the important objects of the theory of automorphic forms, and conversely. We consider classical groups and their unitary, tempered, automorphic and unramified duals. The most important representations in our paper are the isolated points in these duals.