# Homogenous projective factors for actions of semi-simple Lie groups

## Authors

DOI: 10.1007/s002220050377

- Cite this article as:
- Nevo, A. & Zimmer, R. Invent. math. (1999) 138: 229. doi:10.1007/s002220050377

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## 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}*k*λ*dk* 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*.