An orbit model for the spectra of nilpotent Gelfand pairs

TL;DR

This paper develops a geometric model for the spectra of certain nilpotent Gelfand pairs, showing a homeomorphism between spherical functions and orbits in the dual Lie algebra, extending previous results.

Contribution

It establishes a homeomorphism between bounded spherical functions and K-orbits in the dual Lie algebra for a new class of nilpotent Gelfand pairs with specific orbit conditions.

Findings

Proves the homeomorphism for pairs with one-parameter orbit cross sections.

Extends previous results from free and Heisenberg groups.

Supports the conjecture for all nilpotent Gelfand pairs.

Abstract

Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs $(K,N)$ where the $K$-orbits in the center of $N$ have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set $\Delta(K,N)$ of bounded $K$-spherical functions on $N$ and the set $\mathcal{A}(K,N)$ of $K$-orbits in the dual $\mathfrak{n}^*$ of the Lie algebra for $N$ established by Benson and Ratcliff is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for $N$ a free group and $N$ a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs.