Hecke–Bochner identity and eigenfunctions associated to Gelfand pairs on the Heisenberg group

Abstract

Let H n be the ( 2 n + 1 ) -dimensional Heisenberg group, and let K be a compact subgroup of U ( n ) , such that ( K , H n ) is a Gelfand pair. Also assume that the K -action on C n is polar. We prove a Hecke–Bochner identity associated to the Gelfand pair ( K , H n ) . For the special case K = U ( n ) , this was proved by Geller [6] , giving a formula for the Weyl transform of a function f of the type f = P g , where g is a radial function, and P a bigraded solid U ( n ) -harmonic polynomial. Using our general Hecke–Bochner identity we also characterize (under some conditions) joint eigenfunctions of all differential operators on H n that are invariant under the action of K and the left action of H n .