Representation theory and convexity
type: Item
instance of: scholarly article
title: Representation theory and convexity
zbMATH Open document ID: 0964.22004
DOI: 10.1007/BF01234796
author: Karl-Hermann Neeb
published in: Transformation Groups
publication date: 2001-07-09T00:00:00Z
review text: In Section I the essential results on Ol'shanskii semigroups \(S=G \operatorname {Exp} (iW)\), where \(G\) is a Lie group, and invariant convex functions are collected. These semigroups are complex domains generalizing complex reductive groups that show up for \(W=\mathfrak g\) where \(\mathfrak g\) is a compact Lie algebra. Let \(D\subset S\) be a \(G\)-biinvariant domain. Then \(G\times G\) acts on \(D\), and one has to do with harmonic analysis of Hilbert spaces of holomorphic functions \({\mathcal H}\subset Hol(D)\) on which \(G\times G\) acts unitarily. These spaces are called biinvariant Hilbert spaces. In Section II it is shown that irreducible biinvariant Hilbert spaces come from highest weight representations. For example, the unitary representation of \(G\times G\) on a biinvariant Hilbert space \({\mathcal H}\) extends to an action of the semigroup \(S_{\min}\times S_{\min}\) by contractions on \({\mathcal H}\). In Section III the author explains how to construct biinvariant Kähler structures on biinvariant Stein domains and demonstrates with the help of a certain Legendre transform that the studied symplectic manifolds are isomorphic to domains in the cotangent bundle \(T^*(G)\).
reviewed by: A. K. Guts
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 1580014
zbMATH Keywords: Ol'shanskii's semigroup
zbMATH Keywords: Lie group
zbMATH Keywords: representation
zbMATH Keywords: biinvariant Hilbert spaces
zbMATH Keywords: highest weight representations
zbMATH Keywords: biinvariant Kähler structures
zbMATH Keywords: biinvariant Stein domains
zbMATH Keywords: symplectic manifolds
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01234796
OpenAlex ID: W2086191565
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description: scientific article; zbMATH DE number 1580014