Geometric realizations of representations of finite length II

Abstract

Let G = H×sR be a semidirect product Lie group, let O be a locally closed orbit of H in the dual of R, and let S be the subgroup of H stabilizing some point of O. Suppose that U is a representation of length n+1 of G, such that every irreducible representation in the composition series of U is associated to the orbit O and a finite dimensional representation of S by the Mackey machine. We prove that if H is a real linear algebraic group, S is an algebraic subgroup of H, and all finite dimensional representations of S are rational, then U may be realized as a subquotient of the canonical representation of G in the space of functions on the nth-order infinitesimal neighborhood of O in its ambient vector space, taking values in some finite dimensional representation of H.