The orbit method and Gelfand pairs, associated with nilpotent Lie groups

Abstract

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator Open image in new window of the Lie algebra Open image in new window of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.