Deligne-Lusztig varieties and period domains over finite fields

We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig variety which is at the same time a period domain over a finite field. This is done by comparing a cohomology vanishing theorem for DL-varieties, due to Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the first author. We also discuss an affineness criterion for DL-varieties.Comment: 16 pages, reference added to the paper of X. He (arXiv:0707.0259) in which the affineness conjecture for DL-varieties is prove

Wilson Lines from Representations of NQ-Manifolds

An NQ-manifold is a non-negatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual differential geometry, we analyze the subtleties in the generalization to the NQ-setting and we also sketch some possible applications of our construction.Comment: 37 page

An exotic Deligne-Langlands correspondence for symplectic groups

Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules.Comment: v7, 52pages. Corrected typos and errors in the proofs of Lemma 4.1 and Theorem 6.2 modulo Proposition 6.7, final version, accepted for publication in Duke Mat