Handsaw Quiver Varieties and Finite W -algebras

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. 2010 Math. Subj. Class. Primary: 17B37; Secondary: 14D21.