On Some Stratifications of Affine Deligne-lusztig Varieties for Sl3

Let L := k̄((ǫ)), where k is a finite field with q elements and ǫ is an indeterminate, and let σ be the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of σ in L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let f W be the extended affine Weyl group of G. Given x ∈ f W and b ∈ G(L), we have some subgroup of G(L) that acts on the affine Deligne-Lusztig variety Xx(b) = {gI ∈ G(L)/I : gbσ(g) ∈ IxI} and hence a representation of this subgroup on the Borel-Moore homology of the variety. This dissertation investigates this representation for certain b in the cases when G = SL2 and G = SL3.