Abstract.
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 Q⊂G, 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.
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Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999
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Nevo, A., Zimmer, R. Homogenous projective factors for actions of semi-simple Lie groups. Invent. math. 138, 229–252 (1999). https://doi.org/10.1007/s002220050377
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DOI: https://doi.org/10.1007/s002220050377