Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties

Wu, Yaochen

doi:10.1090/ert/650

Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties
HTML articles powered by AMS MathViewer

by Yaochen Wu;

Represent. Theory 27 (2023), 734-765

DOI: https://doi.org/10.1090/ert/650

Published electronically: July 27, 2023

HTML | PDF | Request permission

Abstract:

We give a description of the Namikawa-Weyl group of affinizations of smooth Nakajima quiver varieties based on combinatorial data of the underlying quiver, and compute some explicit examples. This extends a result of McGerty and Nevins for quiver varieties associated to Dynkin quivers.

References

Arnaud Beauville, Symplectic singularities, Invent. Math. 139 (2000), no. 3, 541–549. MR 1738060, DOI 10.1007/s002229900043
Gwyn Bellamy and Alastair Craw, Birational geometry of symplectic quotient singularities, Invent. Math. 222 (2020), no. 2, 399–468. MR 4160872, DOI 10.1007/s00222-020-00972-9
Gwyn Bellamy and Travis Schedler, Symplectic resolutions of quiver varieties, Selecta Math. (N.S.) 27 (2021), no. 3, Paper No. 36, 50. MR 4265941, DOI 10.1007/s00029-021-00647-0
Roman Bezrukavnikov and Ivan Losev, Etingof’s conjecture for quantized quiver varieties, Invent. Math. 223 (2021), no. 3, 1097–1226. MR 4213772, DOI 10.1007/s00222-020-01007-z
Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Ring objects in the equivariant derived Satake category arising from Coulomb branches, Adv. Theor. Math. Phys. 23 (2019), no. 2, 253–344. Appendix by Gus Lonergan. MR 4033353, DOI 10.4310/ATMP.2019.v23.n2.a1
William Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257–293. MR 1834739, DOI 10.1023/A:1017558904030
William Crawley-Boevey, Decomposition of Marsden-Weinstein reductions for representations of quivers, Compositio Math. 130 (2002), no. 2, 225–239. MR 1883820, DOI 10.1023/A:1013793632709
William Crawley-Boevey and Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635. MR 1620538, DOI 10.1215/S0012-7094-98-09218-3
Victor G. Kac, Some remarks on representations of quivers and infinite root systems, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 311–327. MR 607162
Ivan Losev, Isomorphisms of quantizations via quantization of resolutions, Adv. Math. 231 (2012), no. 3-4, 1216–1270. MR 2964603, DOI 10.1016/j.aim.2012.06.017
Ivan Losev, Deformations of symplectic singularities and orbit method for semisimple Lie algebras, Selecta Math. (N.S.) 28 (2022), no. 2, Paper No. 30, 52. MR 4359565, DOI 10.1007/s00029-021-00754-y
I. V. Losev, Symplectic slices for actions of reductive groups, Mat. Sb. 197 (2006), no. 2, 75–86 (Russian, with Russian summary); English transl., Sb. Math. 197 (2006), no. 1-2, 213–224. MR 2230091, DOI 10.1070/SM2006v197n02ABEH003754
Andrea Maffei, A remark on quiver varieties and Weyl groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 3, 649–686. MR 1990675
Kevin McGerty and Thomas Nevins, Kirwan surjectivity for quiver varieties, Invent. Math. 212 (2018), no. 1, 161–187. MR 3773791, DOI 10.1007/s00222-017-0765-x
Kevin McGerty and Thomas Nevins, Springer theory for symplectic Galois groups, Preprint, arXiv:1904.10497, 2019.
Jun Morita, Roots of Kac-Moody Lie algebras, Tsukuba J. Math. 4 (1980), no. 2, 349–354. MR 623447, DOI 10.21099/tkbjm/1496159186
Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, DOI 10.1215/S0012-7094-94-07613-8
Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, DOI 10.1215/S0012-7094-98-09120-7
Yoshinori Namikawa, Poisson deformations of affine symplectic varieties, II, Kyoto J. Math. 50 (2010), no. 4, 727–752. MR 2740692, DOI 10.1215/0023608X-2010-012
Yoshinori Namikawa, Poisson deformations of affine symplectic varieties, Duke Math. J. 156 (2011), no. 1, 51–85. MR 2746388, DOI 10.1215/00127094-2010-066
Arkadij L. Onishchik and Ernest B. Vinberg, Lie groups and algebraic groups, Springer Science & Business Media, 2012.
Peter Slodowy, Four lectures on simple groups and singularities, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, vol. 11, Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, 1980. MR 563725, DOI 10.1007/BFb0090294
Xiuping Su, Flatness for the moment map for representations of quivers, J. Algebra 298 (2006), no. 1, 105–119. MR 2214399, DOI 10.1016/j.jalgebra.2006.01.039