Anosov groups: local mixing, counting, and equidistribution

This is joint work with Samuel Edwards and Hee Oh. Let G be a connected semisimple real algebraic group, and Γ<G be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients ⟨(exptv).f1,f2⟩ in L2(Γ∖G) as t→∞ for any f1,f2∈Cc(Γ∖G) and any vector v in the interior of the limit cone of Γ. These asymptotics involve higher rank analogues of Burger-Roblin measures which will be introduced in this talk. As an application, for an affine symmetric subgroup H of G, we obtain a bisector counting result for Γ-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 Γ-orbits in affine symmetric spaces H∖G.