Global Intersection Cohomology of Quasimaps' Spaces

Abstract

Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$ (Quasiflags), $\CQ^K_\alpha$ (Stable Maps) of $\CQ_\alpha$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $\pi: \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_\alpha,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_\alpha,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincar\'e polynomial'') of the direct sum $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ and construct a natural action of the Lie algebra ${\frak{sl}}_n$ on this direct sum by some middle-dimensional correspondences between Quasiflags' spaces. We conjecture that this module is isomorphic to distributions on nilpotent cone supported at nilpotent subalgebra.