Singular blocks of parabolic category O and finite W-algebras

TL;DR

This paper demonstrates that integral blocks of parabolic category O for semi-simple Lie algebras can be embedded into categories over finite W-algebras, linking algebraic and geometric perspectives.

Contribution

It establishes a new realization of singular blocks of parabolic category O as subcategories of W-algebra categories, extending known equivalences.

Findings

Integral blocks of parabolic category O are realized within finite W-algebra categories.

Singular blocks are geometrically represented as partial Whittaker sheaves.

The work connects algebraic, geometric, and categorical frameworks in representation theory.

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.