Singular blocks of parabolic category O and finite W-algebras

Abstract

We show that each integral infinitesimal block of parabolic category O (including singular ones) for a semi-simple Lie algebra can be realized as a full subcategory of a "thick" category O over a finite W-algebra for the same Lie algebra. The nilpotent used to construct this finite W-algebra is determined by the central character of the block, and the subcategory taken is that killed by a two-sided ideal depending on the original parabolic. The equivalences in question are induced by those of Milicic-Soergel and Losev. We also give a proof of a result of some independent interest: the singular blocks of parabolic category O can be geometrically realized as "partial Whittaker sheaves" on partial flag varieties.

12 pages; v2 and v3: minor corrections suggested by referee; statement of some results changed; v4: additional material added on connection to Slodowy slices and rewrite of introduction

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Cited by in corpus (10)

• 2-block Springer fibers: convolution algebras and coherent sheaves
• Nilpotent orbits and finite W-algebras
• Highest weights for truncated shifted Yangians and product monomial crystals
• A quantum Mirković-Vybornov isomorphism
• Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras
• Quantum Hamiltonian reduction of W-algebras and category O
• Semi-infinite cohomology and the linkage principle for $W$-algebras
• On Singular Localization of $\mathfrak{g}$-modules
• Localization for affine $W$-algebras
• Affine Harish-Chandra bimodules and Steinberg--Whittaker localization