A Geometric Criterion for Gelfand Pairs Associated with Nilpotent Lie Groups

In this paper, we give a proof to the orbit conjecture of Benson–Jenkins–Lipsman–Ratcliff and get a geometric criterion for Gelfand pairs associated with nilpotent Lie groups. Our proof is based on an analysis of the condition by using certain operators naturally attached to two step nilpotent Lie algebras provided with an inner product. A class of nilpotent Lie groups considered by Lauret is also treated as examples.