On Truncation of irreducible representations of Chevalley groups

TL;DR

This paper proves that the dimension of certain cohomology subspaces for arithmetic groups remains bounded regardless of the irreducible representation used, using elementary representation theory of algebraic groups.

Contribution

It establishes a boundedness result for the slope subspace dimensions in cohomology of arithmetic groups, extending the Mazur-Gouvea Conjecture to higher rank Chevalley groups.

Findings

Bounded dimension of slope subspaces in cohomology.

Elementary proof using representation theory of algebraic groups.

Applicable to irreducible modules of Chevalley groups.

Abstract

We prove part of a higher rank analogue of the Mazur-Gouvea Conjecture. More precisely, let $\tilde{\bf G}$ be a connected, reductive ${\Bbb Q}$-split group and let $\Gamma$ be an arithmetic subgroup of $\tilde{\bf G}$. We show that the dimension of the slope $\alpha$ subspace of the cohomology of $\Gamma$ with values in an irreducible $\tilde{\bf G}$-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.