Intersection cohomology of Drinfeld's compactifications
  Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic.   Following Drinfeld, one defines a compactification $\widetilde{\on{Bun}}_P$ of the moduli stack of $P$-bundles on $X$.   The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\on{Bun}}_P$. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$. 
A.Braverman, D.Gaitsgory, M.Finkelberg and I.Mirkovi\'c