Geometric realizations of representations of finite length. II
[Part I in Rev. Math. Phys. 9, 821-851 (1997; Zbl 0881.22011).] Let \(G=H\times_s\mathbb{R}^n\) be a semidirect product Lie group, let \({\mathcal O}\) be a locally closed orbit of \(H\) in the dual of \(\mathbb{R}^n\), and let \(S\) be the subgroup of \(H\) stabilizing some point of \({\mathcal O}\). Suppose that \({\mathcal U}\) is a representation of length \(n+1\) of \(G\), such that every irreducible representation in the composition series of \({\mathcal U}\) is associated to the orbit \({\mathcal O}\) 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 \({\mathcal U}\) may be realized as a subquotient of the canonical representation of \(G\) in the space of functions on the \(n\)th-order infinitesimal neighborhood of \({\mathcal O}\) in its ambient vector space, taking values in some finite-dimensional representation of \(H\).
scientific article; zbMATH DE number 1060829