Some spectral results on ${\displaystyle L^2\left(H_n\right)}$ related to the action of U(p,q)

Let ${\displaystyle H_n}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and ${\displaystyle H_n}$. So ${\displaystyle L^2\left(H_n\right)}$ has a natural structure of G-module. We obtain a decomposition of ${\displaystyle L^2\left(H_n\right)}$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in ${\displaystyle L^2\left(H_n\right)}$ where ${\displaystyle L=\sum _{j=1}^p(X_j^2+Y_j^2)-\sum _{j=p+1}^n(X_j^2+Y_j^2)}$, and where ${\displaystyle \{X_1,Y_1,...,X_n,Y_n,T\}}$ denotes the standard basis of the Lie algebra of ${\displaystyle H_n}$. Finally, we obtain a spectral characterization of the bounded operators on ${\displaystyle L^2\left(H_n\right)}$ that commute with the action of G.