On truncation of irreducible representations of Chevalley groups

We prove part of a higher rank analogue of the Gouvêa–Mazur Conjecture (cf. Gouvêa and Mazur, 1992, [G-M]). More precisely, let G˜ be a connected, reductive QQ-split group and Γ   an arithmetic subgroup of G˜. We show that the dimension of the slope α subspace of the cohomology of Γ   with values in an irreducible G˜-module L is bounded independently of L. The proof is based on general principles of the representation theory of algebraic groups; in particular, we study truncations of highest weight modules of Chevalley groups.