Sheaves on the alcoves and modular representations I

We consider the set of affine alcoves associated with a root system R as a topological space and define a certain category S of sheaves of Zk{mathcal{Z}}_k $$end{document}-modules on this space. Here Zk is the structure algebra of the root system over a field k. To any wall reection s we associate a wall crossing functor on S. In the companion article [FL] we prove that S encodes the simple rational characters of the connected, semisimple and simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.