Dimensions of Some Affine Deligne-lusztig Varieties

Let k be a finite field with q elements, and let k̄ be an algebraic closure of k. We consider the field L := k̄((ǫ)) and its subfield F := k((ǫ)). We write σ : x 7→ x for the Frobenius automorphism of k̄/k, and we also regard σ as an automorphism of L/F in the usual way, so that σ( ∑ anǫ ) = ∑ σ(an)ǫ . We write o for the valuation ring k̄[[ǫ]] of L. Let G be a split connected reductive group over k, and let A be a split maximal torus of G. Put a := X∗(A)R. Write W for the Weyl group of A in G. Fix a Borel subgroup B = AU containing A with unipotent radical U . For λ ∈ X∗(A) we write ǫ for the element of A(F ) obtained as the image of ǫ ∈ Gm(F ) under the homomorphism λ : Gm → A. This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. We begin with the affine Grassmannian. Put K := G(o). We denote by X the affine Grassmannian X = G(L)/K and by x0 its obvious base-point. The group G(L) acts by left translation on X. By the Cartan decomposition G(L) is the disjoint union of the subsets KǫK, with μ running over the dominant elements in X∗(A). For b ∈ G(L) and a dominant coweight μ ∈ X∗(A) the affine Deligne-Lusztig variety Xμ(b) = X G μ (b) is the locally closed subset of X defined by