Regular Orbital Measures on Lie Algebras

TL;DR

This paper investigates the conditions under which the Fourier transform of orbital measures on Lie algebras, supported on Adjoint orbits of regular elements, belongs to L^2(g), revealing a precise dimensional criterion.

Contribution

It establishes a necessary and sufficient condition for the k-th power of the Fourier transform of orbital measures to be in L^2(g), linking it to the dimensions of the Lie algebra and its rank.

Findings

Fourier transform of orbital measures in L^2(g) for k > dim g/(dim g - rank g)

Provides a dimensional criterion for L^2 integrability of Fourier transforms

Enhances understanding of harmonic analysis on Lie algebras

Abstract

Let H be a regular element of an irreducible Lie Algebra g, and let mu be the orbital measure supported on the Adjoint orbit of H. We show that the k-th power of the Fourier transform of mu is in L^2(g) if and only if k > dim g/(dim g-rank g).