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

Schiffmann, Olivier

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

Keywords:

quiver varieties; Kac-Moody algebras; moduli space of sheaves on surfaces