$q$-Hypergeometric polynomials and group-invariant Fourier transformations over a finite field

TL;DR

This paper explores the calculation of matrix elements of group-invariant Fourier transformations over finite fields, expressing them through q-hypergeometric polynomials, and connects these to zonal spherical functions of Gelfand pairs.

Contribution

It introduces a method to compute matrix elements of group-invariant Fourier transforms using q-hypergeometric polynomials and constructs a commutative diagram to facilitate calculations.

Findings

Matrix elements expressed by Krawtchouk and Affine q-Krawtchouk polynomials

Construction of a commutative diagram for transformations

Connection to zonal spherical functions of Gelfand pairs

Abstract

By the Fourier transformations, any group-invariant functions over finite Abelian groups are transformed into group-invariant functions over the character groups. In this paper, we calculate matrix elements of this transformations under specific bases. More specifically, we deal with some vector spaces over a finite field and linear actions. Then the matrix elements under adequate bases are expressed by Krawtchouk or Affine q-Krawtchouk polynomials. For calculations, we construct a commutative diagram which combines two settings of group-invariant Fourier transformations. We apply it to different sized transformations of each example, and solve it inductively. We remark the matrix elements are related to the zonal spherical functions of finite Gelfand pairs.