Invariant measures for horospherical actions and Anosov groups

Abstract

Let $\hat{I}“$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\hat{I}“\backslash G$, up to proportionality, is homeomorphic to ${\mathbb{R}}^{\text{rank}G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger-Roblin measures.

Final version, To appear in IMRN