## Osaka Journal of Mathematics

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

Kentaro Nagao

#### Abstract

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

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

Dates
First available in Project Euclid: 26 October 2009

https://projecteuclid.org/euclid.ojm/1256564211

Mathematical Reviews number (MathSciNet)
MR2583334

Zentralblatt MATH identifier
1246.17020

#### Citation

Nagao, Kentaro. K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials. Osaka J. Math. 46 (2009), no. 3, 877--907. https://projecteuclid.org/euclid.ojm/1256564211

#### References

• J. Beck: Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555--568.
• V. Chari and A. Pressley: Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), 295--326.
• I. Cherednik: A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411--431.
• I. Cherednik: Double affine Hecke algebras and Macdonald's conjectures, Ann. of Math. (2) 141 (1995), 191--216.
• I. Cherednik: Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995), 483--515.
• I. Gordon: Quiver varieties, category $\mathcal{O}$ for rational Cherednik algebras, and Hecke algebras, math.RT/0703150.
• M. Haiman: Combinatorics, symmetric functions, and Hilbert schemes; in Current Developments in Mathematics, 2002, Int. Press, Somerville, MA, 2003, 39--111.
• D. Hernandez: Quantum toroidal algebras and their representations, arXiv: 0801.2397.
• D. Hernandez: Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), 163--200.
• D. Hernandez: Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. (3) 95 (2007), 567--608.
• M. Kashiwara, T. Miwa and E. Stern: Decomposition of $q$-deformed Fock spaces, Selecta Math. (N.S.) 1 (1995), 787--805.
• G. Lusztig: Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365--421.
• G. Lusztig: Remarks on quiver varieties, Duke Math. J. 105 (2000), 239--265.
• I.G. Macdonald: Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki, Vol. 1994/95, Astérisque (1996), Exp. No. 797, 4, 189--207.
• H. Nakajima: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365--416.
• H. Nakajima: Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145--238.
• E.M. Opdam: Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75--121.
• Y. Saito, K. Takemura and D. Uglov: Toroidal actions on level $1$ modules of $U_{q}(\widehat{\mathrm{sl}}_{n})$, Transform. Groups 3 (1998), 75--102.
• O. Schiffmann: The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 2000, 413--440.
• K. Takemura and D. Uglov: Level-$0$ action of $U_{q}(\widehat{\mathfrak{sl}}_{n})$ on the $q$-deformed Fock spaces, Comm. Math. Phys. 190 (1998), 549--583.
• D. Uglov: The trigonometric counterpart of the Haldane Shastry model, hep-th/9508145.
• M. Varagnolo and E. Vasserot: Schur duality in the toroidal setting, Comm. Math. Phys. 182 (1996), 469--483.
• M. Varagnolo and E. Vasserot: Double-loop algebras and the Fock space, Invent. Math. 133 (1998), 133--159.
• M. Varagnolo and E. Vasserot: On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267--297.
• M. Varagnolo and E. Vasserot: On the $K$-theory of the cyclic quiver variety, Internat. Math. Res. Notices 1999, 1005--1028.
• M. Varagnolo and E. Vasserot: Canonical bases and quiver varieties, Represent. Theory 7 (2003), 227--258.