The sup-norm problem for automorphic forms in higher rank

Abstract

We study the sup-norm of Hecke-Maaß cusp forms on certain locally symmetric spaces of SL_n(ℝ). The latter correspond either to cocompact lattices defined by orders in division algebras, or to the Hecke congruence subgroups of SL_n(ℤ), yielding non-compact spaces. The main results are sub-baseline bounds uniform in the volume of the space and, in the compact case, also in the spectral parameter. These bounds are the first of their kind for n > 2. The methods involve a thorough study of level structures in higher rank, including a new reduction theory with level in the non-compact case. This is used to solve the core counting problem by soft, generalisable arguments, based on rigidity principles.