Sheaves on the alcoves and modular representations I

TL;DR

This paper introduces a category of sheaves on affine alcoves linked to a root system, establishing a framework that encodes simple rational characters of algebraic groups over fields with characteristic above the Coxeter number.

Contribution

It defines a sheaf category on affine alcoves and associates wall crossing functors, providing a new geometric approach to modular representation theory.

Findings

Category S encodes simple rational characters

Wall crossing functors relate to root system symmetries

Framework applies to algebraically closed fields with characteristic above Coxeter number

Abstract

We consider the set of affine alcoves associated with a root system R as a topological space and consider a certain category S of sheaves of Z-modules on this space. Here Z is the structure algebra of the root system over a field k. To any wall reflection we associate a wall crossing functor on S. In the companion article "Sheaves on the alcoves and modular representations II" we prove that S encodes the simple rational characters of the connected, simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.