A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs

TL;DR

This paper extends the Frobenius formula to Gelfand pairs, expressing the structure coefficients of their double-class algebras using zonal spherical functions, generalizing classical results for finite groups.

Contribution

It introduces a Frobenius formula for Gelfand pairs, linking structure coefficients to zonal spherical functions, broadening the scope of classical character theory.

Findings

Derived a Frobenius formula for Gelfand pairs.

Expressed structure coefficients in terms of zonal spherical functions.

Generalized classical formulas from finite groups to Gelfand pairs.

Abstract

We generalise some well known properties of irreducible characters of finite groups to zonal spherical functions of Gelfand pairs. This leads to a Frobenius formula for Gelfand pairs. For a given Gelfand pair, the structure coefficients of its associated double-class algebra can be written in terms of zonal spherical functions. This is a generalisation of the Frobenius formula which writes the structure coefficients of the center of a finite group algebra in terms of irreducible characters.