Irreducible components of affine Lusztig varieties

Let G be a loop group and W be its Iwahori-Weyl group. The affine Lusztig variety Yw () describes the intersection of the Bruhat cell I ẇ I for w W with the conjugacy class of G, while the affine Deligne-Lusztig variety Xw (b) describes the intersection of the Bruhat cell I ẇ I with the Frobenius-twisted conjugacy class of b G. Although the geometric connections between these varieties are unknown, numerical relations exist in their geometric properties. This paper explores the irreducible components of affine Lusztig varieties. The centralizer of acts on Yw () and the Frobenius-twisted centralizer of b acts on Xw (b). We relate the number of orbits on the top-dimensional components of Yw () to the numbers of orbits on top-dimensional components of Xw (b) and the affine Springer fibers. For split groups and elements with integral Newton points, we show that, for most w, the numbers of orbits for the affine Lusztig variety and the associated affine Deligne-Lusztig variety match. Moreover, for these, we verify Chi's conjecture that the number of top-dimensional components in Y_ () within the affine Grassmannian equals to the dimension of a specific weight space in a representation of the Langlands dual group.