  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.

Representation Theory;Rings and Algebras