Mixing of all orders of Lie groups actions (Q1187485)

A measure-preserving action of a topological group \(G\) on a probability space \((X,{\mathcal M},\mu)\) is said to be mixing of order \(k\) (or \(k\)- mixing) if for any \(B_ 1,\dots,B_ k\in{\mathcal M}\), and any sequences \(\{g^{(1)}_ i\},\dots,\{g^{(k)}_ i\}\) in \(G\) such that for any \(p\neq q\), \(\{(g^{(p)}_ i)^{-1} g^{(q)}_ i\}\) has no convergent subsequence, we have \(\mu(g^{(1)}_ i B_ 1\cap\cdots\cap g^{(k)}_ i B_ k)\to \mu(B_ 1)\cdots \mu(B_ k)\), as \(i\to\infty\). The author shows that for weakly measurable actions of certain connected Lie groups \(G\) any 2-mixing action is mixing of all orders. This holds in particular if the center of \(G\) is finite and the adjoint group \(\text{Ad }G\) is a closed subgroup of the group of non-singular linear transformations of the Lie algebra of \(G\). The theorem in question is actually stated for all \(G\) with compact center, satisfying the second part of the condition, but the argument given does not hold, if the center is of positive dimension; this observation attributed to A. M. Stepin was communicated to the reviewer by A. N. Starkov; the author proposes to issue an errata in a suitable form. The result shows in particular that any ergodic measure-preserving action of a simple Lie group with finite center is mixing of all orders. A similar assertion also holds for all `irreducible' actions of semisimple Lie groups and yields in particular a result of \textit{B. Marcus} [Invent. Math. 46, 201-209 (1978; Zbl 0395.28012)] for translations of homogeneous spaces of such groups. The present method is notably different from those involved in earlier proofs of higher-order-mixing results and depends on analyzing the invariant measures for actions.