Anosov groups: local mixing, counting and equidistribution

Abstract

For a Zariski dense Anosov subgroup $\Gamma$ of a semisimple real Lie group $G$, we describe the asymptotic behavior of matrix coefficients $\Phi(g)=\langle g f_1, f_2\rangle$ in $L^2(\Gamma\backslash G)$ for local functions $f_1, f_2\in C_c(\Gamma\backslash G)$. These asymptotics involve higher rank analogues of Burger-Roblin measures. As an application, for any symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$. The link between mixing and counting is provided by an equidistribution result for the translates $\Gamma\backslash \Gamma H a$ as $a\to \infty$ in $H\backslash G$.