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

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 $Q$.