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

Abstract

Fix an affine Lie algebra ${\hat{\mathfrak{g}}}_{\kappa }$ with associated principal affine W-algebra ${\mathcal{W}}_{\kappa }$. A basic conjecture of Frenkel--Kac--Wakimoto asserts that Drinfeld--Sokolov reduction sends admissible ${\hat{\mathfrak{g}}}_{\kappa }$-modules to zero or cohomological shifts of minimal series ${\mathcal{W}}_{\kappa }$-modules. We prove this conjecture and a natural generalization to the spectrally flowed Drinfeld--Sokolov reduction functors. This extends the previous results of Arakawa for the minus reduction.

25 pages, comments welcome

References in corpus (2)

• Homological methods in semi-infinite contexts
• Localization for affine $W$-algebras