Harmonic analysis on Heisenberg--Clifford Lie supergroups

TL;DR

This paper develops harmonic analysis tools such as Fourier transform and convolution for Heisenberg--Clifford Lie supergroups, extending classical theorems to this supergeometric setting.

Contribution

It introduces a Fourier transform and convolution product for supergroups, generalizing classical harmonic analysis results to the supergeometry context.

Findings

Defined Fourier transform and convolution on supergroups

Established Fourier transform as an intertwining operator

Generalized classical theorems like Paley--Wiener--Schwartz

Abstract

We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz theorem, and define a convolution Banach algebra.