On the convexity of the moment mapping for unitary highest weight representations
type: Item
instance of: scholarly article
title: On the convexity of the moment mapping for unitary highest weight representations
zbMATH Open document ID: 0829.22012
published in: Journal of Functional Analysis
publication date: 1996-01-17T00:00:00Z
review text: Let \(\pi\) be a unitary representation of a Lie group \(G\) on a Hilbert space \(H\). A vector \(v \in H\) is said to be smooth if the orbit mapping \(G \to H\), \(g \to \pi(g)v\) is smooth. One writes \(H^\infty\) for the space of smooth vectors, and notes that \(d \pi (X)v := (d/dt)|_{t=0} \pi (\text{exp } tX)v\) defines the structure of a \(g\)-module on \(H^\infty\) which extends to a \(g_C\)-module structure via \(d\pi(X + iY) := d\pi(X) + id\pi(Y)\) because \(H^\infty\) is a complex vector space. For each non-zero vector \(v \in H^\infty\) a linear functional on \(g^*\) is defined by \[ \Psi_\pi' : H^\infty \setminus \{0\} \to g^*,\quad \Psi_\pi'(v) (X) := {1\over i} {\langle d\pi(X)v,v \rangle\over \langle v,v\rangle} , \] that is called the moment mapping associated to \(\pi\). To each continuous unitary representation of a Lie group \(G\) in a Hilbert space \(H\) the author associates a moment map from the projective space of smooth vectors \(H^\infty\) to \(g^*\). For unitary highest weight representations one obtains a characterization of those highest weights for which the closure of the image of this map is convex, i.e. equal to the convex hull of the highest weight orbit. This result generalizes the corresponding result for representations of compact groups and holomorphic discrete series representations.
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 763992
zbMATH Keywords: representation
zbMATH Keywords: unitary representation
zbMATH Keywords: moment mapping
zbMATH Keywords: unitary highest weight representations
zbMATH Keywords: representations of compact groups
zbMATH Keywords: holomorphic discrete series representations
author: Karl-Hermann Neeb
reviewed by: A. K. Guts
MaRDI profile type: Publication
full work available at URL: jfan.1995.1014
OpenAlex ID: W1984432755
DOI: 10.1006/JFAN.1995.1014
description: scientific article; zbMATH DE number 763992