  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
$Q$.
