A note on harmonic polynomials on Heisenberg and Carnot groups

We study harmonic polynomial traces on Korányi spheres in the Heisenberg groups and polynomial trace filtrations on general Carnot groups. The homogeneous harmonic trace spaces are algebraically direct and complete on the Heisenberg sphere, but different homogeneous degrees are not generally orthogonal. In fact, on every $\\mathbb H^n$ the degree-one and degree-three spaces fail to be orthogonal for every finite positive full-support surface measure. The decomposition thus uses the orthogonal increments of the cumulative harmonic-degree filtration. These increments give a complete Hilbert sum, have dimension $\\binom{m+2n-1}{2n-1}$, and retain a $U(n)$-equivariant refinement. A constructive triangular recursion produces all homogeneous harmonic polynomials and their dimensions. On an arbitrary Carnot group, the same harmonic filtration always decomposes its closed span; equality with the full spherical $L^2$ space is identified as a separate harmonic-density condition. By contrast, the filtration by all polynomial traces is unconditionally complete by Stone--Weierstrass. We describe its dimensions through the vanishing ideal of the gauge sphere and compute them explicitly for every $\\mathbb H^n$. Our main algebraic result is a Fischer-type decomposition. With $η_+^2(z,t)=|z|^2+4t$, we prove that $P_m(\\mathbb H)=H_m(\\mathbb H)\\oplus η_+^2P_{m-2}(\\mathbb H)$.