Semiinfinite cohomology of quantum groups II

Abstract

It is known that the semi-infinite cohomology spaces of the infinitely twisted nilpotent subalgebra in an affine Lie algebra $g$ with coefficients in an integrable simple module over the affine Lie algebra have a base enumerated by elements of the corresponding affine Weyl group graded by the semiinfinite length function. Let $U$ be the affine quantum group corresponding to $g$. It is possible to define a subalgebra in $U$ being the quantum analogue of the universal enveloping algebra of the infinitely twisted nilpotent subalgebra in $g$. In this paper we prove that for general values of the parameter $v$ the semiinfinite cohomology of this associative algebra with coefficients in an integrable simple module over $U$ coincides with the one of the corresponding Lie subalgebra in $g$ with coefficients in the corresponding $g$-module.