Affine Yangian Action on the Fock Space

Abstract

The localized equivariant homology of the quiver variety of type $A_{N-1}^{(1)}$ can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions, Jack(${\mathfrak{gl}}_N$) symmetric functions introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for ${\mathfrak{sl}}_N$ and the affine Lie algebra ${\widehat{\mathfrak{sl}}}_N$, on both sides. Consequently we obtain affine Yangian action on the Fock space.