Title

An Orbit Model for the Spectra of Nilpotent Gelfand Pairs

Document Type

Article

Publication Date

8-2-2019

Department

Mathematics

Language

English

Publication Title

Transformation Groups

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 L1K(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 Δ(K, N) of bounded K-spherical functions on N and the set A(K, N) of K-orbits in the dual n* of the Lie algebra for N established in [BR08] 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 in [BR08].

Comments

For more information on the published version, visit Springer's Website.

To access a view-only full text published version of this article click https://rdcu.be/b7o6q.

Online access to this article has been shared by the author(s) via Springer Nature SharedIt.

DOI

10.1007/s00031-019-09541-8

Full text currently unavailable.

COinS