Disintegrating tensor representations of nilpotent Lie groups

Abstract

Let G G be a simply connected nilpotent Lie group and H H a closed connected subgroup of G G . Given an irreducible unitary representation π \pi of G G , we present an explicit disintegration of the restriction π | H \pi _{|H} of π \pi to H H . Such a disintegration relies on the description of the double cosets space H ╲ G ╱ B H \diagdown G\diagup B for an arbitrary closed connected subgroup B B of G G , and the well-known smooth disintegration of monomial representations of nilpotent Lie groups. As an application we get a concrete disintegration and a criterion of irreducibility for tensor products of a finite number of irreducible representations of G G .