On Truncation of irreducible representations of Chevalley groups

Abstract

We prove part of a higher rank analogue of the Mazur-Gouvea Conjecture. More precisely, let $\tilde{\mathbf{G}}$ be a connected, reductive $\mathbb{Q}$-split group and let $\Gamma$ be an arithmetic subgroup of $\tilde{\mathbf{G}}$. We show that the dimension of the slope $\alpha$ subspace of the cohomology of $\Gamma$ with values in an irreducible $\tilde{\mathbf{G}}$-module $L$ is bounded independently of $L$. The proof is elementary making only use of general principles of the representation theory of algebraic groups; it is based on consideration of certain truncations of irreducible representations of Chevalley groups.