Inventiones mathematicae

, Volume 138, Issue 2, pp 229-252

First online:

Homogenous projective factors for actions of semi-simple Lie groups

  • Amos NevoAffiliated withTechnion, Israel Institute of Technology, 32000 Haifa, Israel¶ (e-mail:
  • , Robert J. ZimmerAffiliated withDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USA¶ (e-mail:

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We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup QG, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q.

Mathematics Subject Classification (1991): 22D40, 28D15, 47A35, 57S20, 58E40, 60J50