Unitary representation theory of exponential Lie groups
type: Item
instance of: scholarly article
title: Unitary representation theory of exponential Lie groups
zbMATH Open document ID: 0833.22012
author: Horst Leptin
author: Jean Ludwig
published in: De Gruyter Expositions in Mathematics
publication date: 1995-01-25T00:00:00Z
review text: A finite dimensional real Lie group \(G\) is called exponential if the exponential function \(\text{exp}: {\mathfrak g} \to G\) is a diffeomorphism from the Lie algebra \(\mathfrak g\) of \(G\) onto \(G\). It is called solvable if the commutator series \({\mathfrak g},{\mathfrak g}_1 = [{\mathfrak g},{\mathfrak g}]\), \({\mathfrak g}_2 = [{\mathfrak g}_1,{\mathfrak g}_1], \dots\) of \(\mathfrak g\) reaches zero. Bernat showed, generalizing a fundamental result of Kirillov, that the irreducible unitary representations of an exponential solvable Lie group can be parametrized by the \(G\)-orbits in the dual space \({\mathfrak g}^*\) of \(\mathfrak g\) under the coadjoint action. The map \(K : {\mathfrak g}^*/ G \to \widehat {G}\) which associates to a coadjoint orbit the corresponding representation is called the Kirillov map. Both domain and range of the Kirillov map carry natural topologies: On \({\mathfrak g}^*/G\) one has the quotient topology of the Euclidean topology and on \(\widehat{G}\) the induced topology from the projection onto the primitive ideal space of the group \(C^*\)-algebra with its Jacobson topology. The book under review is devoted to the proof of the fact that the Kirillov map is a homeomorphism with respect to these topologies. Kirillov had conjectured this in his 1962 paper for the nilpotent case and I. D. Brown proved it in that case in 1973. The exponential solvable case was settled by J. Ludwig in the late eighties. A crucial ingredient in the proof of Kirillov's conjecture in the exponential case is the possibility to modify the group structure along the way. This allows to write certain positive definite functions in terms of positive definite functions of lower dimensional modified subgroups and use induction to prove the convergence of orbits from the convergence of positive definite functions. Ludwig introduced the concept of variable structures such as \textit{Variable Lie Groups} (VLG) and \textit{Variable Banach Algebras} (VLA) to g…
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22D25
zbMATH DE Number: 715335
zbMATH Keywords: exponential groups
zbMATH Keywords: variable structures
zbMATH Keywords: real Lie group
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: exponential solvable Lie group
zbMATH Keywords: Kirillov map
zbMATH Keywords: Kirillov's conjecture
reviewed by: Joachim Hilgert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 715335