On Some Stratifications of Affine Deligne-Lusztig Varieties for SL_3

Abstract

Let L be k((\epsilon)), where k is an algebraic closure of a finite field with q elements and \epsilon is an indeterminate, and let \sigma be the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of \sigma in L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let W be the extended affine Weyl group of G. Given x in W and b in G(L), we have some subgroup of G(L) that acts on the affine Deligne-Lusztig variety X_x(b) = {gI in G(L)/I : g^{-1}b\sigma(g) is in IxI} and hence a representation of this subgroup on the Borel-Moore homology of the variety. We investigate this representation for certain b in the cases when G is SL_2 and G is SL_3.