Osaka Journal of Mathematics

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

Kentaro Nagao

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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 the nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the quiver varieties.

Article information

Osaka J. Math., Volume 46, Number 3 (2009), 877-907.

First available in Project Euclid: 26 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 33C52: Orthogonal polynomials and functions associated with root systems 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) [See also 32L25, 81Txx] 16G20: Representations of quivers and partially ordered sets


Nagao, Kentaro. K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials. Osaka J. Math. 46 (2009), no. 3, 877--907.

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