Harmonic Analysis of Radon Filtrations for S(n) and GL(n,q)

TL;DR

This paper develops a unified harmonic analysis framework for Radon transforms on symmetric groups and general linear groups over finite fields, enabling explicit computation of spherical functions through recursive decomposition.

Contribution

It introduces a q-analogue approach to Radon transforms, providing a recursive method for decomposing representations and computing spherical functions for these groups.

Findings

Explicit formulas for spherical functions derived

Recursive decomposition of representations achieved

Unified q-analogue framework established

Abstract

We present a unified approach to the study of Radon transforms related to symmetric groups and to general linear groups GL(n,q) regarded as q-analogues of the former. In both cases, we define a sequence of generalized Radon transforms which are intertwining operators for natural representations associated to Gel'fand spaces for our groups. This sequence enables us to decompose in a recursive way these natural representations and to compute explicitly the associated spherical functions. Our methods and results are related by q-analogy.