Sheaves on the alcoves and modular representations II

TL;DR

This paper connects sheaves on alcoves to the representation theory of reductive algebraic groups, showing that indecomposable projectives encode simple rational characters across all characteristics above the Coxeter number.

Contribution

It establishes a link between sheaves on alcoves and modular representations, revealing that indecomposable projectives encode simple characters in all relevant characteristics.

Findings

Indecomposable projective sheaves encode simple rational characters.

The correspondence holds in all characteristics above the Coxeter number.

Provides a geometric framework for understanding modular representations.

Abstract

We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.