Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

Details Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

1997-11-08

arxiv.org/abs/alg-geom/9711009v3

Amer. Math. Soc. Transl. Ser. 2, vol. 194 (1999), 113--148

Mathematics

Algebraic Geometry

version published in AMS Translations

Scientific paper

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis numbered by the irreducible components of certain algebraic cycles, isomorphic to the intersections of semiinfinite orbits in the affine Grassmannian of $G$. We compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$. They carry a natural pure Tate Hodge structure, and their generating functions are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we prove that Kontsevich's space of stable maps provides a natural resolution of $Q$.

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