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 Mosc. Math. J., 2012, Volume 12, Number 3, Pages 633–666 (Mi mmj462)

Handsaw quiver varieties and finite $W$-algebras

Hiraku Nakajima

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Abstract: 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.

Key words and phrases: quiver variety, shifted Yangian, finite $W$-algebra, quantum affine algebra, Kazhdan–Lusztig polynomial.

DOI: https://doi.org/10.17323/1609-4514-2012-12-3-633-666

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MSC: Primary 17B37; Secondary 14D21
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Citation: Hiraku Nakajima, “Handsaw quiver varieties and finite $W$-algebras”, Mosc. Math. J., 12:3 (2012), 633–666

Citation in format AMSBIB
\Bibitem{Nak12} \by Hiraku~Nakajima \paper Handsaw quiver varieties and finite $W$-algebras \jour Mosc. Math.~J. \yr 2012 \vol 12 \issue 3 \pages 633--666 \mathnet{http://mi.mathnet.ru/mmj462} \crossref{https://doi.org/10.17323/1609-4514-2012-12-3-633-666} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3024827} \zmath{https://zbmath.org/?q=an:06126191} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309366400010} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Nawata S., “Givental J-Functions, Quantum Integrable Systems, AGT Relation With Surface Operator”, Adv. Theor. Math. Phys., 19:6 (2015), 1277–1338
2. Yu. Takayama, “Nahm's equations, quiver varieties and parabolic sheaves”, Publ. Res. Inst. Math. Sci., 52:1 (2016), 1–41
3. A. Braverman, M. Finkelberg, H. Nakajima, “Instanton moduli spaces and $\mathcal W$-algebras”, Asterisque, 2016, no. 385, 1–126
4. G. Wilkin, “Moment map flows and the Hecke correspondence for quivers”, Adv. Math., 320 (2017), 730–794
5. Lapa M.F., Turner C., Hughes T.L., Tong D., “Hall Viscosity in the Non-Abelian Quantum Hall Matrix Model”, Phys. Rev. B, 98:7 (2018), 075133