Handsaw quiver varieties and finite W-algebras

Abstract

Following Braverman-Finkelberg-Feigin-Rybnikov (arXiv:1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W-algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang-Mills theory with surface operators. Our new observation is that the C^*-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type A, which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the W-algebra are described in terms of IC sheaves of graded quiver varieties of type A, which were known to be related to Kazhdan-Lusztig polynomials. This gives a new proof of a conjecture by Brundan-Kleshchev on composition multiplicities on Verma modules, which was proved by Losev, in a wider context, by a different method.

31 pages; v.2 The main result was proved earlier by Losev in a wider context by a different method; v.3 typos corrected

References in corpus (2)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• On the structure of the category O for W-algebras

Cited by in corpus (3)

• 3d N=2 mirror symmetry, pq-webs and monopole superpotentials
• ADE Little String Theory on a Riemann Surface (and Triality)
• Givental J-functions, Quantum integrable systems, AGT relation with surface operator