# A structure theorem for actions of semisimple Lie groups

### Abstract

We consider a connected semisimple Lie group $G$ with finite center, an admissible probability measure $\mu$ on $G$, and an ergodic $(G,\mu)$-space $(X,\nu)$. We first note (Lemma 0.1) that $(X,\nu)$ has a unique maximal projective factor of the form $(G/Q,\nu_0)$, where $Q$ is a parabolic subgroup of $G$, and then prove:

1. Theorem 1. If every noncompact simple factor of $G$ has real rank at least two, then the maximal projective factor is nontrivial, unless $\nu$ is a $G$-invariant measure.

2. Theorem 2. For any $G$ of real rank at least two, if the action has positive entropy and fails to have nontrivial projective factor, then $(X,\nu)$ has an equivariant factor space with the same properties, on which $G$ acts via a real-rank-one factor group.

3. Theorem 3. Write $\nu = \nu_0\ast \lambda$, where $\lambda$ is a $P$-invariant measure, $P = MSV$ a minimal parabolic subgroup [F2], [NZ1]. If the entropy $h_\mu(G/P,\nu_0)$ is finite, and every nontrivial element of $S$ is ergodic on $(X,\lambda)$ (or just a well chosen finite set, Theorem 9.1), then ($X,\nu)$ is a measure-preserving extension of its maximal projective factor.

4. The foregoing results are best possible (see Section 11, in particular Theorem 11.4).

We also give some corollaries and applications of the main results. These include an entropy characterization of amenable actions, an explicit entropy criterion for the invariance of $\nu$, and construction of a projective factor for an action of a lattice in $G$ on a compact metric space.

Amos Nevo

Robert J. Zimmer