Quantum Ergodicity on Bruhat-Tits Buildings

Abstract

We study eigenfunctions of the spherical Hecke algebra acting on L^2(Gamma_n backslash G/K) where G = PGL(3, F) with F a non-archimedean local field of characteristic zero, K = PGL(3, O) with O the ring of integers of F, and (Gamma_n) is a sequence of cocompact torsionfree lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself.