A structure theorem for actions of semisimple Lie groups
The article under review considers the ergodic \((G,\mu)\)-spaces \((X,\nu)\) of a connected semisimple Lie group \(G\) with finite center with an admissible probability measure \(\mu \) on \(G\). The authors obtain the following structure theorems for such actions by using the well-known structure theory of semisimple Lie groups and their parabolic subgroups. (1) \((X, \nu)\) has a unique maximal projective factor of the form \((G/ Q, \nu _0)\) for a parabolic subgroup \(Q\) of \(G\). (2) If the simple noncompact factors of \(G\) are of real rank at least two, the maximal projective factor is trivial if and only if the \(\mu\)-stationary measure \(\nu\) is \(G\)-invariant: this result completely characterizes when a \(\mu\)-stationary measure (on an ergodic \(G\)-space) is \(G\)-invariant for such higher rank semisimple Lie groups and it may also be noted that the result is not true without the assumption on the rank of the simple factors. As a corollary to this result an entropy criterion for the existence of a \(G\)-invariant probability measure is given when \(G\) is such a higher rank semisimple Lie group. (3) Suppose the entropy \(h_\mu (X, \nu) = -\int _G \int _X \log {dg^{-1} \nu \over d\nu} (x) d\nu (x) d\mu (g)\) is positive. Then the only obstruction to the existence of a nontrivial projective factor is the existence of a factor group \(G_1\) of \(G\) of real rank one and a nontrivial \((G_1, \mu _1)\)-space \((X_1, \nu _1)\) which is a factor space of \((X, \nu)\) with the same properties. (4) Suppose that \(G\) has real rank at least two. Suppose the entropy of \(\mu\) is finite and the actions of the nontrivial elements of the corresponding maximal \(\mathbb R\)-split torus on \((X, \nu)\) are ergodic. Then \((X, \nu)\) is shown to be a measure preserving extension of its maximal projective factor. As a corollary to this result an entropy criterion is obtained for the amenability of the \((G,\mu)\)-space \((X, \nu)\). As an application of (2) and (3), it is shown that if \(G\) is not simple and the action is irreducible, then \(\nu \) is \(G\)-invariant. The authors also indicate an alternative proof of the intermediate factor theorem proved earlier by the second author.
scientific article; zbMATH DE number 1851143