Localization for affine $W$-algebras

Abstract

We prove a localization theorem for affine $W$-algebras in the spirit of Beilinson--Bernstein and Kashiwara--Tanisaki. More precisely, for any non-critical regular weight $\lambda$, we identify $\lambda$-monodromic Whittaker $D$-modules on the enhanced affine flag variety with a full subcategory of Category $\mathcal{O}$ for the $W$-algebra. To identify the essential image of our functor, we provide a new realization of Category $\mathcal{O}$ for affine $W$-algebras using Iwahori--Whittaker modules for the corresponding Kac--Moody algebra. Using these methods, we also obtain a new proof of Arakawa's character formulae for simple positive energy representations of the $W$-algebra.

33 pages, comments very welcome!

References in corpus (9)

• Vertex Algebras and Algebraic Curves
• Loop group actions on categories and Whittaker invariants
• Quantum Langlands Correspondence
• A Localization Theorem for Finite W-algebras
• Singular blocks of parabolic category O and finite W-algebras
• From $S^1$-fixed points to $\mathcal{W}$-algebra representations
• W-algebras and Whittaker categories
• Affine Beilinson-Bernstein localization at the critical level for $G{L}_2$
• Semi-infinite cohomology and the linkage principle for $W$-algebras

Cited by in corpus (1)

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