Shadows in Coxeter Groups

For a given w in a Coxeter group W, the elements u smaller than w in Bruhat order can be seen as the end alcoves of stammering galleries of type w in the Coxeter complex Σ. We generalize this notion and consider sets of end alcoves of galleries that are positively folded with respect to certain orientation φ of Σ.We call these sets shadows. Positively folded galleries are closely related to the geometric study of affine Deligne–Lusztig varieties, MV polytopes, Hall–Littlewood polynomials, and many more algebraic structures. In this paper, we will introduce various notions of orientations and hence shadows and study some of their algorithmic properties.