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arXiv · 2405.08371

Homogeneous spaces of semidirect products and finite Gelfand pairs

Tullio Ceccherini-Silberstein·Fabio Scarabotti·Filippo Tolli
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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.

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Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli. 2024-05-14. Homogeneous spaces of semidirect products and finite Gelfand pairs. https://arxiv.org/abs/2405.08371

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