Quiver varieties and symmetric pairs

Li, Yiqiang

doi:10.1090/ert/522

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Quiver varieties and symmetric pairs

Author: Yiqiang Li
Journal: Represent. Theory 23 (2019), 1-56
MSC (2010): Primary 16S30, 14J50, 14L35, 51N30, 53D05
DOI: https://doi.org/10.1090/ert/522
Published electronically: January 17, 2019
MathSciNet review: 3900699
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Abstract: We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft-Procesi row/column removal reductions.

[BaK16] M. Balagovic and S. Kolb, Universal K-matrix for quantum symmetric pairs, Journal für die reine und angewandte Mathematik (2016).
[BSWW] Huanchen Bao, Peng Shan, Weiqiang Wang, and Ben Webster, Categorification of quantum symmetric pairs I, Quantum Topol. 9 (2018), no. 4, 643–714. MR 3874000, https://doi.org/10.4171/QT/117
[BW13] Huanchen Bao and Weiqiang Wang, A new approach to Kazhdan-Lusztig theory of type 𝐵 via quantum symmetric pairs, Astérisque 402 (2018), vii+134 (English, with English and French summaries). MR 3864017
[BW16] Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs, Invent. Math. 213 (2018), no. 3, 1099–1177. MR 3842062, https://doi.org/10.1007/s00222-018-0801-5
[BKLW] Huanchen Bao, Jonathan Kujawa, Yiqiang Li, and Weiqiang Wang, Geometric Schur duality of classical type, Transform. Groups 23 (2018), no. 2, 329–389. MR 3805209, https://doi.org/10.1007/s00031-017-9447-4
[BS98] Dan Barbasch and Mark R. Sepanski, Closure ordering and the Kostant-Sekiguchi correspondence, Proc. Amer. Math. Soc. 126 (1998), no. 1, 311–317. MR 1422847, https://doi.org/10.1090/S0002-9939-98-04090-8
[BBD82] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
[BLM] A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of 𝐺𝐿_{𝑛}, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, https://doi.org/10.1215/S0012-7094-90-06124-1
[BGP] I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065
[B03] Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209–216. MR 1996415, https://doi.org/10.1007/s00031-003-0606-4
[BG99] Alexander Braverman and Dennis Gaitsgory, On Ginzburg’s Lagrangian construction of representations of 𝐺𝐿(𝑛), Math. Res. Lett. 6 (1999), no. 2, 195–201. MR 1689209, https://doi.org/10.4310/MRL.1999.v6.n2.a7
[CN18] T.-H. Chen and D. Nadler, Kostant-Sekiguchi homeomorphisms, arXiv:1805.06564.
[Ch84] I. V. Cherednik, Factorizing particles on a half line, and root systems, Teoret. Mat. Fiz. 61 (1984), no. 1, 35–44 (Russian, with English summary). MR 774205
[Ch16] Jaeyoo Choy, Moduli spaces of framed symplectic and orthogonal bundles on ℙ² and the 𝕂-theoretic Nekrasov partition functions, J. Geom. Phys. 106 (2016), 284–304. MR 3508922, https://doi.org/10.1016/j.geomphys.2016.04.011
[BOR15] Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, Rectangular symmetries for coefficients of symmetric functions, Electron. J. Combin. 22 (2015), no. 3, Paper 3.15, 18. MR 3386516
[CG] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
[D] Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197
[DG14] V. Drinfeld and D. Gaitsgory, On a theorem of Braden, Transform. Groups 19 (2014), no. 2, 313–358. MR 3200429, https://doi.org/10.1007/s00031-014-9267-8
[E92] Bas Edixhoven, Néron models and tame ramification, Compositio Math. 81 (1992), no. 3, 291–306. MR 1149171
[ES12] Michael Ehrig and Catharina Stroppel, 2-row Springer fibres and Khovanov diagram algebras for type D, Canad. J. Math. 68 (2016), no. 6, 1285–1333. MR 3563723, https://doi.org/10.4153/CJM-2015-051-4
[ES13] Michael Ehrig and Catharina Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math. 331 (2018), 58–142. MR 3804673, https://doi.org/10.1016/j.aim.2018.01.013
[E09] Naoya Enomoto, A quiver construction of symmetric crystals, Int. Math. Res. Not. IMRN 12 (2009), 2200–2247. MR 2511909, https://doi.org/10.1093/imrn/rnp014
[FRT] N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 193–225. MR 1015339
[Fu03] Baohua Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003), no. 1, 167–186. MR 1943745, https://doi.org/10.1007/s00222-002-0260-9
[G91] Victor Ginzburg, Lagrangian construction of the enveloping algebra 𝑈(𝑠𝑙_{𝑛}), C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 12, 907–912 (English, with French summary). MR 1111326
[GRWa] Nicolas Guay, Vidas Regelskis, and Curtis Wendlandt, Representations of twisted Yangians of types B, C, D: I, Selecta Math. (N.S.) 23 (2017), no. 3, 2071–2156. MR 3663603, https://doi.org/10.1007/s00029-017-0306-x
[GRWb] N. Guay, V. Regelskis, and C. Wendlandt, Representations of twisted Yangians of types B, C, D: II, arXiv:1708.00968.
[H] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
[HL14] Anthony Henderson and Anthony Licata, Diagram automorphisms of quiver varieties, Adv. Math. 267 (2014), 225–276. MR 3269179, https://doi.org/10.1016/j.aim.2014.08.007
[H15] Anthony Henderson, Singularities of nilpotent orbit closures, Rev. Roumaine Math. Pures Appl. 60 (2015), no. 4, 441–469. MR 3436211
[I72] Birger Iversen, A fixed point formula for action of tori on algebraic varieties, Invent. Math. 16 (1972), 229–236. MR 299608, https://doi.org/10.1007/BF01425495
[K14] Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, https://doi.org/10.1016/j.aim.2014.08.010
[KR71] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, https://doi.org/10.2307/2373470
[KP81] Hanspeter Kraft and Claudio Procesi, Minimal singularities in 𝐺𝐿_{𝑛}, Invent. Math. 62 (1981), no. 3, 503–515. MR 604841, https://doi.org/10.1007/BF01394257
[KP82] Hanspeter Kraft and Claudio Procesi, On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982), no. 4, 539–602. MR 694606, https://doi.org/10.1007/BF02565876
[K90] P. B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), no. 2, 473–490. MR 1072915
[KN90] Peter B. Kronheimer and Hiraku Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), no. 2, 263–307. MR 1075769, https://doi.org/10.1007/BF01444534
[Le] Gail Letzter, Harish-Chandra modules for quantum symmetric pairs, Represent. Theory 4 (2000), 64–96. MR 1742961, https://doi.org/10.1090/S1088-4165-00-00087-X
[LW15] Yiqiang Li and Weiqiang Wang, Positivity vs negativity of canonical bases, Bull. Inst. Math. Acad. Sin. (N.S.) 13 (2018), no. 2, 143–198. MR 3792711
[L98] G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141–182. MR 1623674, https://doi.org/10.1006/aima.1998.1729
[L00] George Lusztig, Quiver varieties and Weyl group actions, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 461–489 (English, with English and French summaries). MR 1775358
[L00b] G. Lusztig, Remarks on quiver varieties, Duke Math. J. 105 (2000), no. 2, 239–265. MR 1793612, https://doi.org/10.1215/S0012-7094-00-10523-6
[M02] 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
[M05] Andrea Maffei, Quiver varieties of type A, Comment. Math. Helv. 80 (2005), no. 1, 1–27. MR 2130242, https://doi.org/10.4171/CMH/1
[MO12] D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287.
[M07] Alexander Molev, Yangians and classical Lie algebras, Mathematical Surveys and Monographs, vol. 143, American Mathematical Society, Providence, RI, 2007. MR 2355506
[N94] Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, https://doi.org/10.1215/S0012-7094-94-07613-8
[N96] Hiraku Nakajima, Varieties associated with quivers, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 139–157. MR 1388562
[N98] Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, https://doi.org/10.1215/S0012-7094-98-09120-7
[N00] Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477, https://doi.org/10.1090/S0894-0347-00-00353-2
[N01] Hiraku Nakajima, Quiver varieties and tensor products, Invent. Math. 146 (2001), no. 2, 399–449. MR 1865400, https://doi.org/10.1007/PL00005810
[N03] Hiraku Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), no. 4, 671–721. MR 2023313, https://doi.org/10.1007/s00208-003-0467-0
[N13] Hiraku Nakajima, Quiver varieties and tensor products, II, Symmetries, integrable systems and representations, Springer Proc. Math. Stat., vol. 40, Springer, Heidelberg, 2013, pp. 403–428. MR 3077693, https://doi.org/10.1007/978-1-4471-4863-0_16
[N15] Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \Cal𝑁=4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016), no. 3, 595–669. MR 3565863, https://doi.org/10.4310/ATMP.2016.v20.n3.a4
[N16] Hiraku Nakajima, Lectures on perverse sheaves on instanton moduli spaces, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2017, pp. 381–436. MR 3752464
[N18] Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, https://doi.org/10.1215/S0012-7094-94-07613-8
[O86] Takuya Ohta, The singularities of the closures of nilpotent orbits in certain symmetric pairs, Tohoku Math. J. (2) 38 (1986), no. 3, 441–468. MR 854462, https://doi.org/10.2748/tmj/1178228456
[O91] Takuya Ohta, The closures of nilpotent orbits in the classical symmetric pairs and their singularities, Tohoku Math. J. (2) 43 (1991), no. 2, 161–211. MR 1104427, https://doi.org/10.2748/tmj/1178227492
[OV] A. L. Onishchik and È. B. Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. MR 1064110
[S84] Jir\B{o} Sekiguchi, The nilpotent subvariety of the vector space associated to a symmetric pair, Publ. Res. Inst. Math. Sci. 20 (1984), no. 1, 155–212. MR 736100, https://doi.org/10.2977/prims/1195181836
[S87] Jir\B{o} Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), no. 1, 127–138. MR 867991, https://doi.org/10.2969/jmsj/03910127
[Sl80a] 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
[Sl80b] Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445
[VV00] M. Varagnolo and E. Vasserot, Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), no. 3, 509–533. MR 1885830, https://doi.org/10.1215/S0012-7094-02-11135-1
[VV03] Michela Varagnolo and Eric Vasserot, Canonical bases and quiver varieties, Represent. Theory 7 (2003), 227–258. MR 1990661, https://doi.org/10.1090/S1088-4165-03-00154-7
[Ve95] Michèle Vergne, Instantons et correspondance de Kostant-Sekiguchi, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 8, 901–906 (French, with English and French summaries). MR 1328708
[V86] David A. Vogan Jr., The orbit method and primitive ideals for semisimple Lie algebras, Lie algebras and related topics (Windsor, Ont., 1984) CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 281–316. MR 832204
[V89] David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
[W39] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
[W15] Arik Wilbert, Topology of two-row Springer fibers for the even orthogonal and symplectic group, Trans. Amer. Math. Soc. 370 (2018), no. 4, 2707–2737. MR 3748583, https://doi.org/10.1090/tran/7194