Title

An Orbit Model for the Spectra of Nilpotent Gelfand Pairs

Authors

Holley Friedlander, Dickinson College
William Grodzicki
Wayne Johnson
Gail Radcliff
Anna Romanov
Benjamin Strasser
Brent Wessel

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 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 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

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DOI

10.1007/s00031-019-09541-8

Recommended Citation

Friedlander, Holley, William Grodzicki, Wayne Johnson, Gail Radcliff, Anna Romanov, Benjamin Strasser, and Brent Wessel. "An Orbit Model for the Spectra of Nilpotent Gelfand Pairs."Transformation Groups 25 (2020): 859-886. https://link.springer.com/article/10.1007/s00031-019-09541-8