Lp bounds for higher rank eigenfunctions and asymptotics of spherical functions

Abstract. We prove almost sharp upper bounds for the Lp norms of eigen-functions of the full ring of invariant differential operators on a compact locally symmetric space. Our proof combines techniques from semiclassical analy-sis with harmonic theory on reductive groups, and makes use of asymptotic bounds for spherical functions which improve upon those of Duistermaat, Kolk and Varadarajan and are of independent interest. 1