Intersection Theory and Geometric Dualities on Moduli Spaces of Sheaves

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DMS-0401670Serguei ArkhipovThe project will investigate how semi-infinite cohomology and semi-regular modules integrate into various algebro-geometric constructions on three levels: for finite dimensional objects (including flag varieties, nilpotent cones and Springer varieties of semi-simple algebraic groups), for objects arising in the geometry of loop groups (affine Grassmannians and affine flag varieties) in the course of Geometric Langlands Program due to Beilinson and Drinfeld and, finally, for objects connected with 2-dimensional local fields, e.g. Lie algebras of iterated formal Laurent series with values in finite dimensional Lie algebras. Thus we propose a hierarchy of problems whose unifying ideas are semi-infinite cohomology and semi-regular modules.The proposed project is divided naturally in four parts. In the first part we work out a certain new approach to semi-regular bimodules and semi-infinite cohomology of (nongraded) associative algebras. The second part deals with finite dimensional objects. We describe geometrically the modified semi-regular module over a semi-simple Lie algebra. Then we provide a geometric description for semi-infinite cohomology of the tensor square of the small quantum group at a root of 1. The third part connects semi-infinite cohomology of small quantum groups with perverse sheaves on the affine flag variety, equivariant coherent sheaves on the Springer variety for the Langlands dual group and, finally, with the classical semi-infinite cohomology of the affine Lie algebra at the critical level. The fourth part is an attempt to construct a categorical semi-regular module for the algebra of iterated Laurent series with values in a semi-simple Lie algebra.