Gelfand Pairs Associated with Finite Heisenberg Groups

We examine a family of finite Gelfand pairs which arise in connection with Heisenberg groups H = H n (F) over finite fields of odd characteristic. The symplectic group Sp(n, F) acts on H by automorphisms. A subgroup K of Sp(n, F) yields a Gelfand pair (K, H) when the K-invariant functions on H commute under convolution. This is equivalent to the restriction of the oscillator representation to K being multiplicity free. An interesting example of this type occurs with K a finite analog of the unitary group U(n).