
Journal of Lie Theory 31 (2021), No. 2, 367392 Copyright Heldermann Verlag 2021 Spaces of Bounded Spherical Functions for Irreducible Nilpotent Gelfand Pairs: Part II Chal Benson Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A. bensonf@ecu.edu Gail Ratcliff Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A. ratcliffg@ecu.edu [Abstractpdf] In prior work an orbit method, due to Pukanszky and Lipsman, was used to produce an injective mapping $\Psi\colon \Delta(K,N)\rightarrow\mathfrak{n}^*/K$ 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 $\mathfrak{n}$ of $N$. We have conjectured that $\Psi$ is a topological embedding. In this paper we complete the proof of this conjecture under the hypothesis that $(K,N)$ is an {\it irreducible} nilpotent Gelfand pair. Following Part I of this work it remains to verify the conjecture in six exceptional cases from Vinberg's classification of irreducible nilpotent Gelfand pairs. Keywords: Gelfand pairs, spherical functions, nilpotent Lie groups, orbit method. MSC: 22E30, 43A90. [ Fulltextpdf (226 KB)] for subscribers only. 