On the structure of the category O for W-algebras

Abstract

W-algebra (of finite type) W is a certain associative algebra associated with a semisimple Lie algebra, say g, and its nilpotent element, say e. The goal of this paper is to study the category O for W introduced by Brundan, Goodwin and Kleshchev. We establish an equivalence of this category with certain category of g-modules. In the case when e is of principal Levi type (this is always so when g is of type A) the category of g-modules in interest is the category of generalized Whittaker modules introduced McDowel and studied by Milicic-Soergel and Backelin.

arXiv comment11 pages, v2 some gaps fixed, some proofs rewritten, Remark 5.4 added, v3 15 pages, some gaps fixed, a new section is added

References in corpus (3)

• Representations of shifted Yangians and finite W-algebras
• Harish-Chandra bimodules for quantized Slodowy slices
• Commutative quotients of finite W-algebras

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• Handsaw quiver varieties and finite W-algebras
• Semi-infinite cohomology and the linkage principle for $W$-algebras
• Translation for finite W-algebras
• Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras
• Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras
• On changing highest weight theories for finite W-algebras
• On finite dimensional representations of finite W-superalgebras