On commutative homogeneous vector bundles attached to nilmanifolds

Abstract

The notion of Gelfand pair (G, K) can be generalized if we consider homogeneous vector bundles over G/K instead of the homogeneous space G/K and matrix-valued functions instead of scalar-valued functions. This gives the definition of commutative homogeneous vector bundles. Being a Gelfand pair is a necessary condition of being a commutative homogeneous vector bundle. For the case in which G/K is a nilmanifold having square-integrable representations, in a previous article we determined a big family of commutative homogeneous vector bundles. In this paper, we complete that classification.