Forced gradings and the Humphreys-Verma conjecture

TL;DR

This paper introduces graded analogues of the Humphreys-Verma conjecture for algebraic groups, valid for all prime characteristics, and explores their implications for representation theory and related conjectures.

Contribution

The authors establish the first graded analogues of the Humphreys-Verma conjecture that hold for all prime characteristics, extending previous results limited to large primes.

Findings

Established graded analogues of the Humphreys-Verma conjecture for all p.

Connected the conjecture to finite group representation theory.

Proposed extensions to smaller characteristics based on these results.

Abstract

Let $G$ be a semisimple, simply connected algebraic group defined and split over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a projective indecomposable (rational) $G_r$-module. The well-known Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the $G_r$-action on $Q$ lifts to an rational action of $G$ on $Q$. For $p\geq 2h-2$ (where $h$ is the Coxeter number of $G$), this conjecture was proved by Jantzen in 1980, improving on early work of Ballard. However, it remains open for general characteristics. In this paper, the authors establish several graded analogues of the Humphreys-Verma conjecture, valid for all $p$. The most general of our results, proved in full here, was announced (without proof) in an earlier paper. Another result relates the Humphreys-Verma conjecture to earlier work of Alperin, Collins, and Sibley on finite group representation theory. A key idea in all formulations involves the notion of a forced grading. The latter goes back, in particular, to the recent work of the authors, relating graded structures and $p$-filtrations. The authors anticipate that the Humphreys-Verma conjecture results here will lead to extensions to smaller characteristics of these earlier papers.