We compare the braid group actions on the category of modules over the Lusztig quantum group and on the category of coherent sheaves on the Springer variety. We use this comparison to find semi-infinite cohomology of the small quantum group with coefficients in a quasi-Verma module.

Finally we show that a similar (conjectural) construction for the category of bimodules over the quantum group would provide a precise geometric answer for the semi-infinite cohomology of the square of the small quantum group.