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Dimensions of some affine Deligne-Lusztig varieties. (English) Zbl 1108.14035
The paper under review concerns the dimensions of certain affine Deligne-Lusztig varieties. We first set up the necessary notations.
Let \(L = k((t))\) and \(F = \bar{k}((t))\) for a finite field \(k\). Let \(\sigma\) denote the Frobenius automorphism of \(\bar{k}/k\) and extend it in a natural way to an automorphism of \(F/L\) by demanding \(\sigma(t) = t\). We denote the valuation ring of \(L\) by \({\mathcal O}_L\). Let \(G\) be a split connected reductive \(k\)-group, \(A\) a split maximal torus of \(G\) and \(K = G({\mathcal O}_L) \subseteq G(L)\). For \(b \in G(L)\) and for a dominant \(\mu \in X_*(A)\), the affine Deligne-Lusztig variety is the locally closed subscheme of the affine Grassmannian, \(X := G(L)/K\), defined by \[ X_{\mu}(b) := \{x \in G(L)/K: x^{-1}b\sigma(x) \in K\mu(t)K\} . \] M. Rapoport [Astérisque 298, 271–318 (2005; Zbl 1084.11029)] conjectured a formula for the dimensions of these affine Deligne-Lusztig varieties which was modified later by R. Kottwitz [Pure Appl. Math. Q. 2, No. 3, 817–836 (2006; Zbl 1109.11033)]. Let \(\mathbb D\) be a torus over \(F\) with character group \(\mathbb Q\). The element \(b\) determines a map \(\nu_b : {\mathbb D}\to G\) and the associated coweight \(\bar{\nu}_b \in X_*(A)_{\mathbb Q}\) is dominant. Let \(M_b\) denote the centraliser of \(\nu_b({\mathbb D})\) in \(G\). There exists an inner form \(J\) of \(M_b\) such that \(J(R) = \{g \in G(R \otimes_F L): g^{-1}b\sigma(g) = b\}\) for any \(F\)-algebra \(R\). If \(X_{\mu}(b)\) is nonempty, then the conjectured dimension formula is
\[ \dim X_{\mu}(b) = \langle \rho, \mu-\bar{\nu}_b \rangle-\frac{1}{2}(\text{rk}_F(G)-\text{rk}_F(J)) \] where \(\rho \in X^*(A)_{\mathbb Q}\) denotes the half sum of the positive roots. It is proved in this paper that if this dimension formula holds for “superbasic” elements \(b \in\text{GL}_n(L)\), then it holds in general. This particular case is recently proved by E. Viehmann [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 3, 513–526 (2006; Zbl 1108.14036)] so now the conjecture is completely solved.
The paper also investigates the dimensions of the affine Deligne-Lusztig varieties in the affine flag manifold \(G(L)/I\) where \(I\) is the Iwahori subgroup of \(G(L)\) obtained from an alcove \({\mathbf a}_1\) in the apartment associated to \(A\).

MSC:
14L30 Group actions on varieties or schemes (quotients)
14M15 Grassmannians, Schubert varieties, flag manifolds
20G25 Linear algebraic groups over local fields and their integers
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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