Geometric models for the spectra of certain Gelfand pairs associated with Heisenberg groups

Let K be a compact Lie group acting on a finite dimensional Hermitian vector space V via some unitary representation. Now K acts by automorphisms on the associated Heisenberg group \({H_V=V \times \mathbb{R}}\), and we say that (K, H V ) is a Gelfand pair when the algebra \({L^1_K(H_V)}\) of integrable K-invariant functions on H V commutes under convolution. In this situation an application of the Orbit Method yields a injective mapping \({\Psi}\) from the space Δ(K, H V ) of bounded K-spherical functions on H V to the space \({\mathfrak{h}_V^{*}/K}\) of K-orbits in the dual of the Lie algebra for H V . We prove that \({\Psi}\) is a homeomorphism onto its image provided that the action of K on V is “well-behaved” in a sense made precise in this work. Our result encompasses a widely studied class of examples arising in connection with Hermitian symmetric spaces.