The Shi variety corresponding to an affine Weyl group

Let be an irreducible Weyl group and its affine Weyl group. In this article we show that there exists a bijection between and the integral points of an affine variety, denoted , which we call the Shi variety of . In order to do so, we use Jian‐Yi Shi's characterization of alcoves in affine Weyl groups. We then study this variety further. We introduce a new representation of , called the ‐representation, and we highlight combinatorial and geometrical properties of the irreducible components of via this representation. We also show how the components are related to a fundamental parallelepiped .