Singular blocks of parabolic category O and finite W-algebras

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 modified 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 particular two-sided ideal depending on the original parabolic. The equivalences in question are induced by those defined by Miličić–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.