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 $α\in\BN[I]$ of positive integers one can consider the space $\CQ_α$ of algebraic maps of degree $α$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_α$ (Quasimaps), $\CQ^L_α$ (Quasiflags), $\CQ^K_α$ (Stable Maps) of $\CQ_α$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $π: \CQ^L_α\to \CQ^D_α$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_α,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_α,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincaré polynomial'') of the direct sum $\oplus_{α\in\BN[I]}H^\bullet(\CQ^D_α,IC)$ 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.