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تسجيل مستخدم جديدDimensions of some affine Deligne-Lusztig varieties
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This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties X_mu(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for X_x(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with orbits of the unipotent radical. Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture.
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This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures c
oncerning their dimensions, and generalizes the superset method.
Rapoport and Kottwitz defined the affine Deligne-Lusztig varieties $X_{tilde{w}}^P(bsigma)$ of a quasisplit connected reductive group $G$ over $F = mathbb{F}_q((t))$ for a parahoric subgroup $P$. They asked which pairs $(b, tilde{w})$ give non-empty
varieties, and in these cases what dimensions do these varieties have. This paper answers these questions for $P=I$ an Iwahori subgroup, in the cases $b=1$, $G=SL_2$, $SL_3$, $Sp_4$. This information is used to get a formula for the dimensions of the $X_{tilde{w}}^K(sigma)$ (all shown to be non-empty by Rapoport and Kottwitz) for the above $G$ that supports a general conjecture of Rapoport. Here $K$ is a special maximal compact subgroup.
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Let F be a non-archimedean local field, let L be the maximal unramified extension of F, and let fr be the Frobenius automorphism. Let G be a split connected reductive group over F, and let B(1) be the Bruhat-Tits building associated to G(F). We know
that fr acts on G(L) with fixed points G(F). Let I be the Iwahori associated to a chamber in B(1). We have the relative position map, inv, from G(L)/I x G(L)/I to the extended affine Weyl group, W_e of G. If w is in W_e and b is in G(L), then the affine Deligne-Lusztig set Xw(b fr) is {x in G(L)/I : inv(x,b fr(x)) = w}. This paper answers the question of which Xw(b fr) are non-empty for certain G and b.
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