K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials

Details K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials

2007-09-12

arxiv.org/abs/0709.1767v1

Osaka J. Math. 46 (2009), no. 3, 877--907

Mathematics

Quantum Algebra

28 pages

Scientific paper

We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $\hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q-Fock space using nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the quiver varieties.

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