Abstract Harmonic Analysis on the General Linear Group GL(n,R)

TL;DR

This paper extends harmonic analysis to the general linear group by generalizing Fourier transforms and proving the Plancherel theorem, considering the group's disconnected structure and the isomorphism between its components.

Contribution

It introduces a generalized Fourier transform for GL(n,R) and establishes the Plancherel theorem, accounting for the group's disconnected components and their isomorphism.

Findings

Generalized Fourier transform for GL(n,R)

Plancherel theorem for the general linear group

Isomorphism between components with positive and negative determinants

Abstract

Consider the general linear group, which is not connected but rather has two connected components, the matrices with positive determinant and the ones with negative determinant. Consider the Iwasawa decomposition of its special linear group. We adopt the technique of the paper [12] to generalize the definition of the Fourier transform and to obtain the Plancherel theorem for the special linear group. Besides we prove that the component with negative determinant has a structure of group, which is isomorphic onto the group of the component with positive determinant in order to obtain the Plancherel theorem the general linear group