Algebraic Geometry Seminar

This is a continuation of the talk given the previous week. The Springer correspondence relates nilpotent orbits in the Lie algebra of a reductive algebraic group to irreducible representations of the Weyl group. We develop a Springer theory in the case of symmetric spaces using Fourier transform, which relates nilpotent orbits in this setting to irreducible representations of Hecke algebras of various Coexter groups with speci ed parameters. A key ingredient in the construction is the nearby cycle sheaves associated to the adjoint quotient map. We also make use of Vogan duality, a strengthening of Langlands duality for real groups. Our Springer correspondence also involves Hessenberg varieties. Simple examples of such varieties include classical objects in algebraic geometry: Jacobians, Fano varieties of k-planes in the intersection of two quadrics, etc. We explain a general strategy for computing the cohomology of Hessengerg varieties and illustrate the method by computing the cohomology of Fano varieties of linear subspaces in the intersection of two quadrics. The talks are partly based on joint work with Tsao-hsien Chen and with Misha Grinberg.