The Laumon's resolution of Drinfeld's compactification is small

Abstract

Let $C$ be a smooth projective curve of genus 0. Let $\FF$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha$ of positive integers one can consider the space $\MM\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\FF$. This space has drawn much attention recently in connection with Quantum Cohomology. The space $\MM\alpha$ is smooth but not compact. The problem of compactification of $\MM\alpha$ proved very important. One compactification $\MML\alpha$ (the space of {\em quasiflags}), was constructed in \cite{L}. However, historically the first and most economical compactification $\MMD\alpha$ (the space of {\em quasimaps}) was constructed by Drinfeld (early 80-s, unpublished). The latter compactification is singular, while the former one is smooth. Drinfeld has conjectured that the natural map $\pi:\MML\alpha\to\MMD\alpha$ is a small resolution of singularities. In the present note we prove this conjecture. As a byproduct, we compute the stalks of $IC$ sheaf on $\MMD\alpha$ and, moreover, the Hodge structure in these stalks. Namely, the Hodge structure is a pure Tate one, and the generating function for the $IC$ stalks is just the Lusztig's $q$-analogue of Kostant's partition function (see \cite{Lu}).