Modules over the small quantum group and semi-infinite flag manifold
  We develop a theory of perverse sheaves on the semi-infinite flag manifold $G((t))/N((t))\cdot T[[t]]$, and show that the subcategory of Iwahori-monodromy perverse sheaves is equivalent to the regular block of the category of representations of the small quantum group at an even root of unity. 
S. Arkhipov, R. Bezrukavnikov, A. Braverman, D. Gaitsgory, I.   Mirkovi\'c