A structure theorem for actions of semisimple Lie groups
type: Item
instance of: scholarly article
title: A structure theorem for actions of semisimple Lie groups
zbMATH Open document ID: 1012.22038
DOI: 10.2307/3597198
author: Amos Nevo
author: Robert J. Zimmer
published in: Annals of Mathematics. Second Series
publication date: 2003-06-13T00:00:00Z
review text: The article under review considers the ergodic \((G,\mu)\)-spaces \((X,\nu)\) of a connected semisimple Lie group \(G\) with finite center with an admissible probability measure \(\mu \) on \(G\). The authors obtain the following structure theorems for such actions by using the well-known structure theory of semisimple Lie groups and their parabolic subgroups. (1) \((X, \nu)\) has a unique maximal projective factor of the form \((G/ Q, \nu _0)\) for a parabolic subgroup \(Q\) of \(G\). (2) If the simple noncompact factors of \(G\) are of real rank at least two, the maximal projective factor is trivial if and only if the \(\mu\)-stationary measure \(\nu\) is \(G\)-invariant: this result completely characterizes when a \(\mu\)-stationary measure (on an ergodic \(G\)-space) is \(G\)-invariant for such higher rank semisimple Lie groups and it may also be noted that the result is not true without the assumption on the rank of the simple factors. As a corollary to this result an entropy criterion for the existence of a \(G\)-invariant probability measure is given when \(G\) is such a higher rank semisimple Lie group. (3) Suppose the entropy \(h_\mu (X, \nu) = -\int _G \int _X \log {dg^{-1} \nu \over d\nu} (x) d\nu (x) d\mu (g)\) is positive. Then the only obstruction to the existence of a nontrivial projective factor is the existence of a factor group \(G_1\) of \(G\) of real rank one and a nontrivial \((G_1, \mu _1)\)-space \((X_1, \nu _1)\) which is a factor space of \((X, \nu)\) with the same properties. (4) Suppose that \(G\) has real rank at least two. Suppose the entropy of \(\mu\) is finite and the actions of the nontrivial elements of the corresponding maximal \(\mathbb R\)-split torus on \((X, \nu)\) are ergodic. Then \((X, \nu)\) is shown to be a measure preserving extension of its maximal projective factor. As a corollary to this result an entropy criterion is obtained for the amenability of the \((G,\mu)\)-space \((X, \nu)\). As an application of (2) and (3), …
Mathematics Subject Classification ID: 22F10
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 58E40
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 37A35
zbMATH DE Number: 1851143
zbMATH Keywords: semisimple Lie groups
zbMATH Keywords: ergodic spaces
zbMATH Keywords: projective factor
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: entropy
Wikidata QID: Q115228661
reviewed by: C. R. E. Raja
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/3597198
OpenAlex ID: W1977736572
description: scientific article; zbMATH DE number 1851143