Abstract Harmonic Analysis On the non Singular Matrix Lie Group

TL;DR

This paper advances the understanding of harmonic analysis on non-connected Lie groups by defining Fourier transforms, establishing Plancherel theorem, and classifying ideals on a specific semi-direct product group.

Contribution

It introduces a Fourier-Mellin transform and proves the Plancherel theorem for a non-connected solvable Lie group, addressing a longstanding problem in the field.

Findings

Established Fourier-Mellin transform on group G

Proved Plancherel theorem for G

Classified all left ideals of the group algebra

Abstract

As well known that it is no way to do the abstract harmonic analysis on the non connected Lie groups. The goal of this paper is to draw the attention of Mathematicians to solve this problem. therefore let R be the group of nonzero real numbers with multiplication and let H be the 3-dimensional Heisenberg group. I denote by G the semi direct of the Heisenberg group with R3 which is non connected solvable Lie group, and isomorphic onto the 6- dimensional non singular triangular matrix Lie group. I define the Fourier-Mellin transform and establish the Plancherel theorem on the group G. Besides I prove the solvability of any non zero invariant differential operator on the identity component of the group G. Finally, I give a classification of all left ideals of the its group algebra