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

C. Benson ; G. Ratcliff

Journal of Lie theory, Tome 31 (2021) no. 2, pp. 367-392

Résumé

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.

Détail

Classification : 22E30, 43A90
Mots-clés : Gelfand pairs, spherical functions, nilpotent Lie groups, orbit method