Gelfand pairs associated with finite Heisenberg groups

A topological group G together with a compact subgroup K are said to form a Gelfand pair if the set L1(K\G/K) of K-bi-invariant integrable functions on G is a commutative algebra under convolution. The situation where G and K are Lie groups has been the focus of extensive and ongoing investigation. Riemannian symmetric spaces G/K furnish the most widely studied and best understood examples. ([Hel84] is a standard reference.) Apart from these, key examples arise as semi-direct products G = K n N , of compact Lie groups K with two-step nilpotent Lie groups N . Such pairs are the focus of [BJR90], [Vin03] and [Yak06], among other works. There are many examples where N = Hn(R), a (real) Heisenberg group, and K is a subgroup of the unitary group U(n). Gelfand pairs also arise in connection with analysis on finite groups, but, to our knowledge, have been studied less extensively. Known examples include the symmetric group modulo the hyperoctahedral group [Mac] and finite analogues of the hyperbolic plane [SA87, Ter99]. In this paper we introduce a family of Gelfand pairs associated with finite Heisenberg groups. They provide finite analogues for the Gelfand pairs associated with Hn(R). Our examples appear elsewhere, in the literature on the oscillator representation, but their relevance to the study of Gelfand pairs has not, however, been previously emphasized.