  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.
