The Fourier transform on 2-step Lie groups

TL;DR

This paper develops a simplified Fourier transform framework for 2-step Lie groups, analyzing singular frequencies and degeneracies, extending previous results to a broader class of groups.

Contribution

It introduces a new, more manageable Fourier transform approach for 2-step Lie groups and studies the structure of singular frequencies and degeneracies within this context.

Findings

Simplified Fourier transform framework for 2-step Lie groups

Analysis of singular frequencies and degeneracies

Extension of previous results to broader group classes

Abstract

In this paper, we investigate the behavior of the Fourier transform on finite dimensional 2-step Lie groups and develop a general theory akin to that of the whole space or the torus. We provide a familiar framework in which computations are made sensibly easier than with the usual representation-theoretic Fourier transform. In addition, we study the 'singular frequencies' of the group, at which the canonical bilinear antisymmetric form degenerates. We also exhibit a specific example for which partial degeneracy of the canonical form occurs, as opposed to the full degeneracy at the origin. We thus extend the results from [1].