Sheaves on the alcoves I: Projectivity and wall crossing functors

This paper is the first in a series of papers in which we define and study a category of “sheaves of $\mathcal{Z}$-modules on the set of alcoves” that carries important information on the category of representations of semisimple Lie algebras in positive characteristics. Here, we define this category and study the structure of its projective objects using a new version of wall crossing functors.