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Invariant measures on G /Γ for split simple Lie groups G
Author(s) -
Einsiedler Manfred,
Katok Anatole
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10092
Subject(s) - mathematics , lie group , simple lie group , pure mathematics , haar measure , invariant (physics) , entropy (arrow of time) , combinatorics , mathematical physics , physics , quantum mechanics
We study the left action α of a Cartan subgroup on the space X = G /Γ, where Γ is a lattice in a simple split connected Lie group G of rank n > 1. Let μ be an α‐invariant measure on X . We give several conditions using entropy and conditional measures, each of which characterizes the Haar measure on X . Furthermore, we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure. The main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures. © 2003 Wiley Periodicals, Inc.