Boundary Distributions for GL3 over a Local Field andSymmetric Power Coefficients

Abstract

In this thesis, we construct a residue map and a Poisson kernel between holomorphic discrete series representations on the Drinfeld period domain and harmonic cocycles with certain non-trivial coefficients on the Bruhat-Tits building for GL3 over a local field of any characteristic. In order to construct the Poisson kernel, we find a new locally analytic kernel function that can be integrated against general boundary distributions. Assuming the existence of certain boundary distributions attached to bounded harmonic \ncocycles, we prove that the Poisson kernel is a right inverse of the residue map for bounded harmonic cocycles. Moreover, we show that the existence of the needed boundary distributions follows from a non-criticality statement for a new class of automorphic forms. We prove a control theorem that implies this non-criticality statement for trivial coefficients. Finally, we apply our constructions to relate spaces of Drinfeld cusp forms for certain congruence subgroups of GL3 and spaces of harmonic cocycles extending work of Teitelbaum to GL3.