Homogeneous spaces of semidirect products and finite Gelfand pairs

Let K≤H be two finite groups and let C≤A be two finite abelian groups, with H acting on A as a group of automorphisms admitting C as a K-invariant subgroup. We study the homogeneous space X:=H⋉A/K⋉C and determine the decomposition of the permutation representation of H⋉A acting on X. We then characterize when this is multiplicity-free, that is, when H⋉A,K⋉C 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.