We show that stationary characters on irreducible lattices \(\Gamma < G\) of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice \(\Gamma < G\), the left regular representation \(\lambda _{\Gamma }\) is weakly contained in any weakly mixing representation \(\pi \). We prove that for any such irreducible lattice \(\Gamma < G\), any Uniformly Recurrent Subgroup (URS) of \(\Gamma \) is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices \(\Gamma < G\). The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

我们证明了高阶连通半单李群的不可约格\(\Gamma < G\)上的平稳特征是共轭不变的，也就是说，它们是真特征。该结果在表示论、算子代数、遍历理论和拓扑动力学中有多种应用。特别地，我们证明对于任何这样的不可约格\(\Gamma < G\)，左正则表示\(\lambda _{\Gamma }\)弱地包含在任何弱混合表示\(\pi \)中。我们证明，对于任何这样的不可约格\(\Gamma < G\) ， \(\Gamma \)的任何均匀循环子群 (URS)都是有限的，回答了 Glasner-Weiss 的问题。我们还获得了不可约格格\(\Gamma < G\)的 Peterson 字符刚性结果的新证明。我们论文的主要新颖之处是冯·诺依曼代数上格的平稳作用的结构定理。