Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology

We find a counterpart of the classical fact that the regular representation R(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G(0) subset of G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g circle plus g-module. The standard regular representation R((G) over cap) of the affine group (G) over cap has two commuting actions of the Lie algebra (g) over cap ith total central charge 0, and carries the structure of a conformal field theory. The modified versions R'(G) and R'((G) over cap (0)), originating from the loop version of the Bruhat decomposition, have two commuting (g) over cap -actions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G = SL(2, C). The quantum Drinfeld-Sokolov reduction transforms the representations of the affine Lie algebras into their W-algebra counterparts, and can be used to produce analogues of the modified regular representations. When g=sl(2, C) the corresponding W-algebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations, which are vertex operator algebras with the total central charge equal to 26.