Geometric realizations of representations of finite length. II
type: Item
instance of: scholarly article
title: Geometric realizations of representations of finite length. II
zbMATH Open document ID: 0921.22010
author: Charles H. Conley
published in: Pacific Journal of Mathematics
publication date: 1997-09-16T00:00:00Z
review text: [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 \(H\).
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 1060829
zbMATH Keywords: Lie group
zbMATH Keywords: locally closed orbit
zbMATH Keywords: representation
zbMATH Keywords: composition series
zbMATH Keywords: Mackey machine
MaRDI profile type: Publication
full work available at URL: pjm.1998.183.201
OpenAlex ID: W2081966570
DOI: 10.2140/PJM.1998.183.201
description: scientific article; zbMATH DE number 1060829