Orientation of Alcoves in Affine Weyl Groups

Abstract Let $W$ be an irreducible Weyl group and $W_{a}$ its affine Weyl group. In [3], the author introduced an affine variety $\widehat{X}_{W_{a}}$, called the Shi variety of $W_{a}$, whose integral points are in bijection with $W_{a}$. The set of irreducible components of $\widehat{X}_{W_{a}}$ provided results at the intersection of group theory, combinatorics, and geometry. In this article, we express the notion of orientation of alcoves in terms of the first cohomology group of $W$ and the irreducible components of the Shi variety. We also provide modular equations in terms of Shi coefficients that describe explicitly the property of having the same orientation.