Spaces of Bounded Spherical Functions on Heisenberg Groups: Part I

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 HV = V × R yields a Gelfand pair. In previous work we have applied the Orbit Method to produce an injective mapping Ψ from the space ∆(K,HV ) of bounded K-spherical functions on HV to the space h ∗ V /K of K-orbits in the dual of the Lie algebra for HV . We have shown that Ψ 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 Ψ : ∆(K,HV )→ hV /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.