A New Spherical Harmonics on the Heisenberg Group

TL;DR

This paper introduces a novel set of spherical harmonics tailored for the Heisenberg group, highlighting their orthogonal polynomial properties and potential applications in harmonic analysis.

Contribution

It presents a new class of spherical harmonics specifically designed for the Heisenberg group, expanding the tools available for analysis on this non-commutative structure.

Findings

Development of new spherical harmonics for the Heisenberg group

Establishment of orthogonal polynomial properties of these harmonics

Potential applications in harmonic analysis and representation theory

Abstract

Let $\h_n$ be the $(2n+1)$-dimensional Heisenberg group. and let ${\cal L}_\alpha$ be the sublaplacian of the Lie algebra of $\h_n$ A new spherical harmonics with its orthogonal polynomial properties is presented for the group.