Singular blocks of parabolic category O and finite W-algebras

Details Singular blocks of parabolic category O and finite W-algebras Singular blocks of parabolic category O and finite W-algebras

2009-09-10

arxiv.org/abs/0909.1860v4

Mathematics

Representation Theory

12 pages; v2 and v3: minor corrections suggested by referee; statement of some results changed; v4: additional material added

Scientific paper

10.1016/j.jpaa.2011.03.020

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.

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