Exotic symmetric space over a finite field, I |
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Authors: | T Shoji K Sorlin |
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Institution: | 1. Department of Mathematics, Tongji University, 2139 Siping Road, Shanghai, 200092, P.R. China 2. L.A.M.F.A, CNRS UMR 7352, Université de Picardie-Jules Verne, 33 rue Saint Leu, F-80039, Amiens, Cedex I, France
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Abstract: | Let V be a 2n-dimensional vector space over an algebraically closed field k with ch k ≠ 2. Let G = GL(V) and H = Sp2n be the symplectic group obtained as H = G θ for an involution θ on G. We also denote by θ the induced involution on $ \mathfrak{g} $ = Lie G. Consider the variety G/H × V on which H acts naturally. Let $ \mathfrak{g}_{\mathrm{nil}}^{{-\theta }} $ be the set of nilpotent elements in the -1 eigenspace of θ in $ \mathfrak{g} $ . The role of the unipotent variety for G in our setup is played by $ \mathfrak{g}_{\mathrm{nil}}^{{-\theta }} $ × V, which coincides with Kato’s exotic nilpotent cone. Kato established, in the case where k = C, the Springer correspondence between the set of irreducible representations of the Weyl group of type C n and the set of H-orbits in $ \mathfrak{g}_{\mathrm{nil}}^{{-\theta }} $ × V by applying Ginzburg theory for affine Hecke algebras. In this paper we develop a theory of character sheaves on G/H × V, and give an alternate proof for Kato’s result on the Springer correspondence based on the theory of character sheaves. |
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