Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces

Michael Chow, Pratyush Sarkar

Published 2021-05-24Version 1

Let $G$ be a connected semisimple real algebraic group and $\Gamma < G$ be an Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp(t\mathsf{v}) : t \in \mathbb{R}\}$ on $\Gamma \backslash G$ in any interior direction $\mathsf{v}$ of the limit cone of $\Gamma$ with respect to the Bowen--Margulis--Sullivan measure associated to $\mathsf{v}$. When $\Gamma$ is the fundamental group of a compact negatively curved manifold, this was proved earlier by Sambarino for $M$-invariant functions, where $M$ is the centralizer of the flow. By the work of Edwards--Lee--Oh which is a higher rank extension of Roblin's transverse intersection argument, an immediate application is an asymptotic formula for matrix coefficients in $L^2(\Gamma \backslash G)$.