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Nakajima’s quiver varieties (after Nakajima, Lusztig, Varagnolo, Vasserot, Crawley-Boevey, et al.). (Variétés carquois de Nakajima (d’après Nakajima, Lusztig, Varagnolo, Vasserot, Crawley-Boevey, et al.).) (French) Zbl 1151.14026
Séminaire Bourbaki. Volume 2006/2007. Exposés 967–981. Paris: Société Mathématique de France (ISBN 978-2-85629-253-2/pbk). Astérisque 317, 295-344, Exp. No. 976 (2008).
Summary: Motivated by the study of the moduli space of vector bundles on certain complex surfaces (the so-called “ALE” spaces), Nakajima introduced in the early 1990s a new class of symplectic algebraic varieties $$\mathfrak{M}_Q$$ associated to any finite quiver $$Q$$. He realizes in the cohomology of these varieties $$\mathfrak {M}_Q$$ (as well as in the cohomology of certain Lagrangian subvarieties $$\mathfrak{L}_Q$$ of $$\mathfrak{M}_Q$$) all the highest weight irreducible integrable representations of the Kac-Moody algebra corresponding to $$Q$$. He also obtains a geometric realization of the crystals (in the sense of Lusztig and Kashiwara) of these representations. This yields an analogue (for Kac-Moody algebras and quantum groups) of the theory developed by Kazhdan-Lusztig and Ginzburg in order to describe the Langlands correspondence for $$p$$-adic groups and affine Hecke algebras.
For the entire collection see [Zbl 1151.00015].

##### MSC:
 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 16G20 Representations of quivers and partially ordered sets
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