On the Affineness of Deligne-lusztig Varieties

We prove that the Deligne-Lusztig variety associated to minimal length elements in any δ-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [10]. 1.1 Notations. Let k be an algebraic closure of the finite prime field Fp and G be a connected reductive algebraic group over k with an endomorphism F : G → G such that some power F d of F is the Frobenius endomorphism relative to a rational structure over a finite subfield k0 of k. Let q be the positive number with q d = |k0|. We fix a F -stable Borel subgroup B and a F -stable maximal torus T ⊂ B. Let Φ be the set of roots and (αi)i∈I be the set of simple roots corresponding to (B, T ). For i ∈ I, let ø∨i be the corresponding fundamental coweight. Let W = N(T )/T be the Weyl group and (si)i∈I be the set of simple reflections. For w ∈ W , let l(w) be the length of w. Since (B, T ) is F -stable, F induces a bijection on I and an automorphism on W . We denote the induced maps on W and I by δ. Now δ also induces isomorphisms on the set of characters X = Hom(T,Gm) and the set of cocharacters X ∨ = Hom(Gm, T ) which we also denote by δ. Then it is easy to see that F μ = q · δı(μ) for μ ∈ X. For J ⊂ I, let ΦJ be the set of roots generated by {αj}j∈J and WJ be the subgroup of W generated by {sj}j∈J . Let W J be the set of minimal length coset representatives for W/WJ . The unique maximal element in W will be denoted by w0 and the unique maximal element in WJ will be denoted by w J 0 . Let α0 = ∑ i∈I niαi be the highest root and n0 = ∑ i∈I ni. 1.2. Let B be the set of Borel subgroups of G. For w ∈ W , let O(w) = {(gB, B); g ∈ G} be the G-orbit on B × B that corresponding to w. Set X(w) = {B ∈ B; (B, F (B)) ∈ O(w)}. This is the Deligne-Lusztig variety associated to w (see [2, 1.4]). It is known that X(w) is a variety of pure dimension l(w) (see loc.cit.) and 2000 Mathematics Subject Classification. 20C33, 20F55. The author is partially supported by NSF grant DMS-0700589.