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Characterization of the Gelfand pair: Quasi Gelfand pair

Cornelie Mitcha Malanda

Abstract

Let G be a locally compact group and let K be a compact subgroup of Aut(G), the group of automorphisms of G. The pair $(G, K )$ is a Gelfand pair if the algebra $L^{1}_{K}(G)$ of K-invariant integrable functions on G is commutative under convolution. In \cite{toure2008lie}, the charactezations of this algebra in the nilpotent case were studied, which generalize some results obtained by C. Benson, J. Jenkins, G. Ratcliff in \cite{benson1990gel} and obtained a new criterion for Gelfand pairs. In this paper we describe the spherical function associated with this type of pair.

Characterization of the Gelfand pair: Quasi Gelfand pair

Abstract

Let G be a locally compact group and let K be a compact subgroup of Aut(G), the group of automorphisms of G. The pair is a Gelfand pair if the algebra of K-invariant integrable functions on G is commutative under convolution. In \cite{toure2008lie}, the charactezations of this algebra in the nilpotent case were studied, which generalize some results obtained by C. Benson, J. Jenkins, G. Ratcliff in \cite{benson1990gel} and obtained a new criterion for Gelfand pairs. In this paper we describe the spherical function associated with this type of pair.
Paper Structure (7 sections, 4 theorems, 30 equations)

This paper contains 7 sections, 4 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Quasi Gelfand pair
  3. Quasi Spherical function
  4. Comment on the Gelfand transform associted:
  5. Examples
  6. Comment:
  7. Acknowledgements

Key Result

Theorem 2.1

If $(G , K)$ is a quasi-Gelfand pair then $G$ is unimodular.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • proof