A note on harmonic polynomials on Heisenberg and Carnot groups

TL;DR

This paper extends the classical spherical harmonic decomposition to the setting of the Heisenberg and Carnot groups, revealing orthogonal decompositions of harmonic polynomials and their traces on spheres.

Contribution

It introduces a harmonic polynomial decomposition on the Heisenberg group and general Carnot groups, generalizing Euclidean spherical harmonics to these non-commutative settings.

Findings

Decomposition of $L^2$ spaces into finite-dimensional harmonic subspaces.

Unique polynomial decomposition involving the gauge function.

Extension of spherical harmonic theory to Carnot groups.

Abstract

In this paper, we consider homogeneous $\Delta_H$-harmonic polynomials on the first Heisenberg group $\mathbb H$ and their traces on the unit sphere $S_\rho$ associated with the Kor\'anyi--Folland homogeneous norm $\rho$. We prove that $L^2(S_\rho,\sigma)$ decomposes as the orthogonal Hilbert direct sum of finite-dimensional spaces $H_m(S_\rho)$ of spherical harmonics of degree $m$, in direct analogy with the classical Euclidean spherical harmonic decomposition. We also show that, for the polynomial gauge $\eta_+^2(z,t)=|z|^2+4t$, every homogeneous polynomial on $\mathbb H$ admits a unique decomposition $$ P_m(\mathbb H) = H_m(\mathbb H)\oplus \eta_+^2 P_{m-2}(\mathbb H). $$ Finally, we extend the spherical $L^2$-decomposition to general Carnot groups $G$ equipped with a canonical homogeneous norm $N$ associated with a fundamental solution of a fixed sub-Laplacian $\Delta_G$. The traces on $S_N$ of homogeneous $\Delta_G$-harmonic polynomials of degree $m$ form pairwise orthogonal eigenspaces of the spherical operator on $S_N$, and their span is dense in $L^2(S_N,\sigma_N)$.