A note on harmonic polynomials on Heisenberg and Carnot groups

In this paper, we consider homogeneous ${\mathrm{\Delta }}_H$-harmonic polynomials on the first Heisenberg group $\mathbb{H}$ and their traces on the unit sphere $S_{\rho }$ associated with the Korányi–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 ${\eta }_{+}^2(z,t)=\mathrm{\mid }z{\mathrm{\mid }}^2+4t$, every homogeneous polynomial on $\mathbb{H}$ admits a unique decomposition $$P_m(\mathbb{H})=H_m(\mathbb{H})\oplus {\eta }_{+}^2{P}_{m-2}(\mathbb{H})\mathrm{.}$$ 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 ${\mathrm{\Delta }}_G$. The traces on $S_N$ of homogeneous ${\mathrm{\Delta }}_G$-harmonic polynomials of degree $m$ form pairwise orthogonal eigenspaces of the spherical operator on $S_N$, and their span is dense in $L^2(S_N,{\sigma }_N)$.