The Steinberg quotient of a tilting character

TL;DR

This paper investigates the properties of Steinberg quotients of tilting characters in simple algebraic groups over fields of prime characteristic, revealing well-behaved multiplicities in these quotients for indecomposable tilting modules.

Contribution

It introduces the concept of Steinberg quotients of tilting characters and analyzes their multiplicities, providing new insights into their structure in modular representation theory.

Findings

Multiplicities of orbit sums in Steinberg quotients are well behaved for indecomposable tilting modules.

Steinberg quotients are divisible by the Steinberg character in the context of projective modules over Frobenius kernels.

The structure of these quotients offers new understanding of tilting modules in algebraic groups.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field of prime characteristic. If $M$ is a finite dimensional $G$-module that is projective over the Frobenius kernel of $G$, then its character is divisible by the character of the Steinberg module. In this paper we study such quotients, showing that if $M$ is an indecomposable tilting module, then the multiplicities of the orbit sums appearing in its "Steinberg quotient" are well behaved.