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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $A$ 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.
- M. Balagovic and S. Kolb, Universal K-matrix for quantum symmetric pairs, Journal für die reine und angewandte Mathematik (2016).
- 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, DOI https://doi.org/10.4171/QT/117
- 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
- Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs, Invent. Math. 213 (2018), no. 3, 1099–1177. MR 3842062, DOI https://doi.org/10.1007/s00222-018-0801-5
- 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, DOI https://doi.org/10.1007/s00031-017-9447-4
- 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, DOI https://doi.org/10.1090/S0002-9939-98-04090-8
- 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
- A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of ${\rm GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI https://doi.org/10.1215/S0012-7094-90-06124-1
- 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
- Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209–216. MR 1996415, DOI https://doi.org/10.1007/s00031-003-0606-4
- Alexander Braverman and Dennis Gaitsgory, On Ginzburg’s Lagrangian construction of representations of ${\rm GL}(n)$, Math. Res. Lett. 6 (1999), no. 2, 195–201. MR 1689209, DOI https://doi.org/10.4310/MRL.1999.v6.n2.a7
- T.-H. Chen and D. Nadler, Kostant-Sekiguchi homeomorphisms, arXiv:1805.06564.
- 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
- Jaeyoo Choy, Moduli spaces of framed symplectic and orthogonal bundles on $\Bbb {P}^2$ and the $K$-theoretic Nekrasov partition functions, J. Geom. Phys. 106 (2016), 284–304. MR 3508922, DOI https://doi.org/10.1016/j.geomphys.2016.04.011
- 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, DOI https://doi.org/10.37236/4808
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
- Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197
- V. Drinfeld and D. Gaitsgory, On a theorem of Braden, Transform. Groups 19 (2014), no. 2, 313–358. MR 3200429, DOI https://doi.org/10.1007/s00031-014-9267-8
- Bas Edixhoven, Néron models and tame ramification, Compositio Math. 81 (1992), no. 3, 291–306. MR 1149171
- 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, DOI https://doi.org/10.4153/CJM-2015-051-4
- Michael Ehrig and Catharina Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math. 331 (2018), 58–142. MR 3804673, DOI https://doi.org/10.1016/j.aim.2018.01.013
- Naoya Enomoto, A quiver construction of symmetric crystals, Int. Math. Res. Not. IMRN 12 (2009), 2200–2247. MR 2511909, DOI https://doi.org/10.1093/imrn/rnp014
- 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
- Baohua Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003), no. 1, 167–186. MR 1943745, DOI https://doi.org/10.1007/s00222-002-0260-9
- Victor Ginzburg, Lagrangian construction of the enveloping algebra $U({\rm sl}_n)$, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 12, 907–912 (English, with French summary). MR 1111326
- 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, DOI https://doi.org/10.1007/s00029-017-0306-x
- N. Guay, V. Regelskis, and C. Wendlandt, Representations of twisted Yangians of types B, C, D: II, arXiv:1708.00968.
- 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
- Anthony Henderson and Anthony Licata, Diagram automorphisms of quiver varieties, Adv. Math. 267 (2014), 225–276. MR 3269179, DOI https://doi.org/10.1016/j.aim.2014.08.007
- Anthony Henderson, Singularities of nilpotent orbit closures, Rev. Roumaine Math. Pures Appl. 60 (2015), no. 4, 441–469. MR 3436211
- Birger Iversen, A fixed point formula for action of tori on algebraic varieties, Invent. Math. 16 (1972), 229–236. MR 299608, DOI https://doi.org/10.1007/BF01425495
- Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI https://doi.org/10.1016/j.aim.2014.08.010
- B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, DOI https://doi.org/10.2307/2373470
- Hanspeter Kraft and Claudio Procesi, Minimal singularities in ${\rm GL}_{n}$, Invent. Math. 62 (1981), no. 3, 503–515. MR 604841, DOI https://doi.org/10.1007/BF01394257
- 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, DOI https://doi.org/10.1007/BF02565876
- P. B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), no. 2, 473–490. MR 1072915
- Peter B. Kronheimer and Hiraku Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), no. 2, 263–307. MR 1075769, DOI https://doi.org/10.1007/BF01444534
- Gail Letzter, Harish-Chandra modules for quantum symmetric pairs, Represent. Theory 4 (2000), 64–96. MR 1742961, DOI https://doi.org/10.1090/S1088-4165-00-00087-X
- 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
- G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141–182. MR 1623674, DOI https://doi.org/10.1006/aima.1998.1729
- 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
- G. Lusztig, Remarks on quiver varieties, Duke Math. J. 105 (2000), no. 2, 239–265. MR 1793612, DOI https://doi.org/10.1215/S0012-7094-00-10523-6
- 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
- Andrea Maffei, Quiver varieties of type A, Comment. Math. Helv. 80 (2005), no. 1, 1–27. MR 2130242, DOI https://doi.org/10.4171/CMH/1
- D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287.
- Alexander Molev, Yangians and classical Lie algebras, Mathematical Surveys and Monographs, vol. 143, American Mathematical Society, Providence, RI, 2007. MR 2355506
- Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, DOI https://doi.org/10.1215/S0012-7094-94-07613-8
- 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
- Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, DOI https://doi.org/10.1215/S0012-7094-98-09120-7
- Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477, DOI https://doi.org/10.1090/S0894-0347-00-00353-2
- Hiraku Nakajima, Quiver varieties and tensor products, Invent. Math. 146 (2001), no. 2, 399–449. MR 1865400, DOI https://doi.org/10.1007/PL00005810
- Hiraku Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), no. 4, 671–721. MR 2023313, DOI https://doi.org/10.1007/s00208-003-0467-0
- 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, DOI https://doi.org/10.1007/978-1-4471-4863-0_16
- Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal N=4$ gauge theories, I, Adv. Theor. Math. Phys. 20 (2016), no. 3, 595–669. MR 3565863, DOI https://doi.org/10.4310/ATMP.2016.v20.n3.a4
- 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
- Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, DOI https://doi.org/10.1215/S0012-7094-94-07613-8
- 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, DOI https://doi.org/10.2748/tmj/1178228456
- 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, DOI https://doi.org/10.2748/tmj/1178227492
- 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
- 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, DOI https://doi.org/10.2977/prims/1195181836
- 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, DOI https://doi.org/10.2969/jmsj/03910127
- 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
- Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445
- M. Varagnolo and E. Vasserot, Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), no. 3, 509–533. MR 1885830, DOI https://doi.org/10.1215/S0012-7094-02-11135-1
- Michela Varagnolo and Eric Vasserot, Canonical bases and quiver varieties, Represent. Theory 7 (2003), 227–258. MR 1990661, DOI https://doi.org/10.1090/S1088-4165-03-00154-7
- 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
- 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, DOI https://doi.org/10.1007/bf01394418
- 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
- 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
- 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, DOI https://doi.org/10.1090/tran/7194
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Additional Information
Yiqiang Li
Affiliation:
Department of Mathematics, University at Buffalo, the State University of New York, Buffalo, New York 14260
MR Author ID:
828279
ORCID:
0000-0003-4608-3465
Email:
yiqiang@buffalo.edu
Keywords:
Nakajima variety,
partial Springer resolution,
nilpotent Slodowy slices of classical groups,
rectangular symmetry,
column/row removal reduction,
symmetric pairs,
$\mathscr {K}$-matrix
Received by editor(s):
January 15, 2018
Received by editor(s) in revised form:
October 15, 2018, and November 2, 2018
Published electronically:
January 17, 2019
Additional Notes:
This work was partially supported by the National Science Foundation under the grant DMS 1801915.
Article copyright:
© Copyright 2019
American Mathematical Society