Gelfand pairs attached to representations of compact Lie groups

Abstract

For each compact Lie algebra g and each real representationV of g we construct a two-step nilpotent Lie groupN(g, V), endowed with a natural left-invariant riemannian metric. The main goal of this paper is to show that this construction produces many new Gelfand pairs associated with nilpotent Lie groups. Indeed, we will give a full classification of the manifoldsN(g, V) which are commutative spaces, using a characterization in terms of multiplicity-free actions.