Bounded K-spherical functions on Heisenberg groups

Abstract

Let H_n be the (2n + 1)-dimensional Heisenberg group, and let K be a compact subgroup of Aut(H_n), the group of automorphisms of H_n. The pair (K, H_n) is called a Gelfand pair if L_K¹(H_n), the subalgebra of elements of L¹(H_n) that are invariant under the action of K, is commutative. In this case, the continuous homomorphisms on L_K¹(H_n) are given by integrating against certain K-invariant functions on H_n. These functions are the K-spherical functions associated to the Gelfand pair (K, H_n). In this paper we show how to compute the bounded K-spherical functions on H_n.