Spectral Synthesis on the Reduced Heisenberg Group

The spectral synthesis problem for the phase space $\mathbb{C}^n$ associated with the reduced Heisenberg group $H^n_{\rm{red}}$ is studied. The paper deals with the case of subspaces in $\mathcal{E}(\mathbb{C}^n)$ invariant under the twisted shifts
$$ f(z)\to f(z-w)e^{(i/2)\operatorname{Im}\langle z,{w}\rangle},\qquad w\in\mathbb{C}^n, $$
and the action of the unitary group $U(n)$. It is shown that any such subspace is generated by the root vectors of a special Hermite operator contained in this subspace. As a corollary, we obtain the spectral synthesis theorem for subspaces in $\mathcal{E}(H^n_{\rm{red}})$ invariant under the unilateral shifts and the action of the unitary group $U(n)$.