The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras

Let g be a semisimple Lie algebra. The affine W-algebra associated to g is a topological algebra which quantizes the algebraic loop space of the Kostant slice. It is constructed as a quantum Hamiltonian (alias quantum Drinfeld--Sokolov) reduction of the corresponding affine Lie algebra. The admissible representations (resp., minimal series) are a family of highest weight modules for affine Lie algebras (resp., affine W-algebras) distinguished by their role in rational conformal field theory. A basic pair of conjectures proposed by Frenkel--Kac--Wakimoto state that two versions of the Drinfeld--Sokolov reduction functor send admissible representations of affine Lie algebras to minimal series representations of W-algebras. For one version, the 'minus' reduction, this was settled by Arakawa. We have confirmed the conjecture for the outstanding 'plus' reduction. The main new technical inputs come from the local quantum geometric Langlands program.