Spaces of bounded spherical functions on Heisenberg groups: part I

Abstract

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 Heisenberg group $$H_V=V\times \mathbb {R}$$ yields a Gelfand pair. The condition that $$K:V$$ be “well-behaved” establishes a relationship between the associated moment mapping and highest weight vectors occurring in the polynomial ring $${\mathbb {C}}[V]$$ . Under this condition, an application of the Orbit Method produces a topological embedding of the space of bounded spherical functions for $$(K,H_V)$$ in the space of $$K$$ -orbits in the dual of the Lie algebra for $$H_V$$ . In part I of this work, it was shown that every irreducible multiplicity free action is well-behaved. Here we extend this result to encompass all multiplicity free actions. Our proof uses case-by-case analysis of multiplicity free actions which are indecomposable but not irreducible.