A simple character formula

TL;DR

This paper proves a character formula for simple representations of certain algebraic groups, replacing Lusztig's conjecture under specific characteristic conditions, advancing understanding in modular representation theory.

Contribution

It introduces a new character formula for simple modules in the principal block, valid when the characteristic exceeds a certain bound, offering an alternative to Lusztig's conjecture.

Findings

Character formula expressed via baby Verma modules

Valid for characteristic greater than 2h-1

Provides a practical replacement for Lusztig's conjecture

Abstract

In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group G in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than 2h-1, where h is the Coxeter number of G. This provides a replacement for Lusztig's conjecture, valid under a reasonable assumption on the characteristic.