Gelfand Pairs with Coherent States

Abstract

The notion of solvable Gelfand pairs (K,N) (K is a compact Lie group acting on N, a solvable connected and simply connected Lie group) is due to Benson, Jenkins and Ratcliff. Thanks to the localization lemma, they came back to the case where K is a connected subgroup of U(n) acting on N = Hn, the 2n + 1-dimensional Heisenberg group. They gave a geometrical condition for such a pair: (K,Hn) is a Gelfand pair if and only if the intersection of each coadjoint orbit of G = K ⊳ Hn with (Lie K)⊥ contains at most one integral K-orbit. Using coherent states, we define here a generating function of multiplicity mρ for each ρ in K^. mρ is holomorphic on D(0,1), mρ (r) = ∑n = 0∞ an rn, an ∈ℕ and limr → 1 mρ (r) = mtp (ρ, Wν) (Wν is the generic representation of Hn naturally extended to K). (K,Hn) is thus a Gelfand pair if and only if limr → 1 mρ 1. We prove here that if mρ is a non homogeneous function, then there is at least two K-orbits in the intersection of the generic coadjoint orbit associated to ρ with (Lie K)⊥.