Plancherel Theorem and the Left Ideals of the Group Algebra for the Jacobi Group

TL;DR

This paper develops a Fourier transform framework for the Jacobi group, a semidirect product involving the Heisenberg group, and classifies its left ideals to establish a Plancherel theorem relevant to quantum mechanics.

Contribution

It introduces a Fourier transform for the Jacobi group and classifies all left ideals of its group algebra, extending harmonic analysis to this important group.

Findings

Established the Plancherel theorem for the Jacobi group

Classified all left ideals of the group algebra for the Jacobi group

Provided a foundation for harmonic analysis in quantum mechanics contexts

Abstract

Let G be the three dimensional connected real semisimple Lie group and let KAN be the Iwasawa decomposition of G.Let J be the Jacobi group, which is the semidirect product of the two groups Heisenberg group with G. The Jacobi group plays an important role in Quantum Mechanics. The purpose of this paper is to define the Fourier transform in order to obtain the Plancherel theorem for the group J. To this end a classification of all left ideals of the group algebra for the Jacobi group