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 $\mathbb{C}^*$-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.