Let G = H ×sℝn be a
semidirect product Lie group, let be a locally closed orbit of H in the dual of ℝn,
and let S be the subgroup of H stabilizing some point of . Suppose that 𝒰 is
a of length n + 1 of G, such that every irreducible representation in the
composition series of 𝒰 is associated to the orbit and a finite dimensional 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 s of S are
rational, then 𝒰 may be realized as a subquotient of the canonical of G
in the space of functions on the nth-order infinitesimal neighborhood of
in its ambient vector space, taking values in some finite dimensional of
H.
