Spaces of bounded spherical functions on Heisenberg groups: part II

Consider a linear multiplicity free action by a compact Lie group \(K\) on a finite dimensional hermitian vector space \(V\). Letting \(K\) act on the associated Heisenberg group, \(H_V=V\times \mathbb{R }\) yields a Gelfand pair. In previous work, we have applied the Orbit Method to produce an injective mapping \(\Psi \) from the space \(\Delta (K,H_V)\) 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 have shown that \(\Psi \) is a homeomorphism onto its image provided that \(K:V\) is a “well-behaved” multiplicity free action. In this paper, we prove that \(K:V\) is well-behaved whenever \(K\) acts irreducibly on \(V\). Thus, if \(K:V\) is an irreducible multiplicity free action then \(\Psi :\Delta (K,H_V)\rightarrow \mathfrak{h }_V^*/K\) is a homeomorphism onto its image. Our proof involves case-by-case analysis working from the classification of irreducible multiplicity free actions. A sequel to this paper will extend these results to encompass non-irreducible actions.