Ben Webster
Published 2009-09-10, updated 2011-01-27Version 4
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.
Comments: 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
Cohomology in singular blocks of parabolic category $\mathcal{O}$
Radon transforms of twisted D-modules on partial flag varieties
Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
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