Spaces of Bounded Spherical Functions for Irreducible Nilpotent Gelfand Pairs: Part I

In prior work an orbit method, due to Pukanszky and Lipsman, was used to produce an injective mapping Psi : Delta (K, N) n*/ K from the space of bounded K-spherical functions for a nilpotent Gelfand pair (K, N) into the space of K-orbits in the dual for the Lie algebra n of N. We have conjectured that Psi is a topological embedding. This has been proved for all pairs (K, N) with N a Heisenberg group. A nilpotent Gelfand pair (K, N) is said to be irreducible if K acts irreducibly on n/ [n, n]. In this paper and its sequel we will prove that Psi is an embedding for all such irreducible pairs. Our proof involves careful study of the non-Heisenberg entries in Vinberg's classification of irreducible nilpotent Gelfand pairs. Part I concerns generalities and six related families of examples from Vinberg's list in which the center for n can have arbitrarily large dimension.