Homogeneous spaces of semidirect products and finite Gelfand pairs

TL;DR

This paper investigates the structure of certain homogeneous spaces formed by semidirect products of finite groups, characterizes when they form Gelfand pairs, and computes associated spherical functions, connecting to Dunkl's work on Johnson schemes.

Contribution

It provides a general framework for analyzing homogeneous spaces of semidirect products and characterizes when these pairs are Gelfand pairs, including explicit spherical functions.

Findings

Decomposition of permutation representations on the homogeneous space.

Criteria for the pair to be a Gelfand pair (multiplicity-free).

Explicit calculation of spherical functions in the Gelfand pair case.

Abstract

Let $K\leq H$ be two finite groups and let $C\leq A$ be two finite abelian groups, with $H$ acting on $A$ as a group of isomorphisms admitting $C$ as a $K$-invariant subgroup. We study the homogeneous space $X\coloneqq\left(H\ltimes A\right)/\left(K\ltimes C\right)$ and determine the decomposition of the permutation representation of $H\ltimes A$ acting on $X$. We then characterize when this is multiplicity-free, that is, when $\left(H\ltimes A,K\ltimes C\right)$ is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl's results on the $q$-analog of the nonbinary Johnson scheme.