Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 21 (2026), 61 -- 78
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This work is licensed under a Creative Commons Attribution 4.0 International License.

SUGGESTIONS TO STUDY AFFINE AND GIT QUOTIENTS OF THE EXTENDED GROTHENDIECK--SPRINGER RESOLUTION

Mee Seong Im

Abstract. We define filtered ADHM data and connect a notion of filtered quiver representations to Grothendieck--Springer resolutions. We also provide current developments and give a list of research problems to further study filtered ADHM equation.

2020 Mathematics Subject Classification: 16G20; 20G05; 20G20; 14C05.
Keywords: Nakajima quiver variety; geometric invariant theory; nonreductive group; filtered ADHM data; Hilbert scheme; rational Cherednik algebra

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References

  1. W. Borho, R. MacPherson, Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707--710. MR618892. Zbl 0467.20036. DOI: https://cir.nii.ac.jp/crid/1573387449508494592.
  2. N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser, Boston, 2010. x+495 pp. MR2838836. Zbl 1185.22001. DOI: https://doi.org/ 10.1007/978-0-8176-4938-8.
  3. A. Craw, Quiver flag varieties and multigraded linear series, Duke Math. J. 156 (2011), no. 3, 469--500. MR2772068. Zbl 1213.14026. DOI: https://doi.org/ 10.1215/00127094-2010-217.
  4. W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compos. Math. 126 (2001), no. 3, 257--293. MR1834739. Zbl 1037.16007. DOI: https://doi.org/10.1023/A:1017558904030.
  5. W. Crawley-Boevey, Lectures on Representations of Quivers, 1992, 1--38. rlhttps://www.math.uni-bielefeld.de/~wcrawley/quivlecs.pdf.
  6. H. Derksen, J. Weyman, Semi-invariants of quivers and saturation for Littlewood--Richardson coefficients, J. Amer. Math. Soc. 13 (2000), no. 3, 467--479. MR1758750. Zbl 0993.16011. DOI: https://doi.org/10.1090/S0894-0347-00-00331-3.
  7. M. Domokos, A. N. Zubkov, Semi-invariants of quivers as determinants, Transform. Groups 6 (2001), no. 1, 9--24. MR1825166. Zbl 0984.16023. DOI: https://doi.org/10.1007/BF01236060.
  8. S. Donkin, Invariants of several matrices, Invent. Math. 110 (1992), no. 2, 389--401. MR1185589. Zbl 0826.20036. DOI: https://doi.org/10.1007/BF01231338.
  9. S. Donkin, Polynomial invariants of representations of quivers, Comment. Math. Helv. 69 (1994), no. 1, 137--141. MR1259609. Zbl 0816.16015. DOI: https://doi.org/10.1007/BF02564477.
  10. W. L. Gan, V. Ginzburg, Almost-commuting variety, \mathscrD-modules, and Cherednik algebras, Int. Math. Res. Pap. 2006 (2006), Article ID 26439, 1--54. MR2210660. Zbl 1158.14006. DOI: https://doi.org/10.1155/imrp/2006/26439.
  11. V. Ginzburg, Geometric methods in the representation theory of Hecke algebras and quantum groups, in Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514 (1998), 127--183. MR1649626. Zbl 1009.17011. DOI: https://doi.org/10.1007/978-94-015-9131-7_4.
  12. V. Ginzburg, Lectures on Nakajima's quiver varieties, 2009, 1--40. rlhttp://arxiv.org/pdf/0905.0686v2.
  13. F. D. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Math., 1673, Springer-Verlag, Berlin, 1997. vi+148 pp. MR1489234. Zbl 0886.14020. DOI: https://doi.org/10.1007/BFb0093525.
  14. M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941--1006. MR1839919. Zbl 1009.14001. DOI: https://doi.org/10.1090/S0894-0347-01-00373-3.
  15. M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), no. 2, 371--407. MR1918676. Zbl 1053.14005. DOI: https://doi.org/10.1007/s002220200219.
  16. M.S. Im, On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck--Springer resolution, Ph.D. thesis, University of Illinois at Urbana--Champaign, ProQuest LLC, Ann Arbor, MI, 2014. vi+127. MR3312842. Zbl 1607.02209. DOI: https://hdl.handle.net/2142/49392.
  17. M.S. Im, Semi-invariants of filtered quiver representations with at most two pathways, Contemp. Math. 835 (2026), 215--229. MR5023831. Zbl 1409.0702. DOI: https://doi.org/10.1090/conm/835.
  18. M.S. Im, The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck--Springer resolution, J. Geom. Phys. 132 (2018), 84--98. MR3836769. Zbl 1403.14081. DOI: https://doi.org/10.1016/j.geomphys. 2018.05.022.
  19. M.S. Im, T. Scrimshaw, The regularity of almost-commuting partial Grothendieck--Springer resolutions and parabolic analogs of Calogero--Moser varieties, J. Lie Theory 31 (2021), no. 1, 127--148. MR4161536. Zbl 1469.14100. DOI: https://www.heldermann.de/JLT/JLT31/JLT311/jlt31007.htm.
  20. M. Kashiwara, R. Rouquier, Microlocalization of rational Cherednik algebras, Duke Math. J. 144 (2008), no. 3, 525--573. MR2444305. Zbl 1147.14002. DOI: https://doi.org/10.1215/00127094-2008-043.
  21. M. Khovanov, A. D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309--347. MR2525917. Zbl 1188.81117. DOI: https://doi.org/10.1090/S1088-4165-09-00346-X.
  22. M. Khovanov, A. D. Lauda, A diagrammatic approach to categorification of quantum groups. II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685--2700. MR2763732. Zbl 1214.81113. DOI: https://doi.org/10.1090/S0002-9947-2010-05210-9.
  23. L. Le Bruyn, C. Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), no. 2, 585--598. MR958897. Zbl 0693.16018. DOI: https://doi.org/10.1090/S0002-9947-1990-0958897-0.
  24. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447--498. MR1035415. Zbl 0703.17008. DOI: https://doi.org/10.1090/S0894-0347-1990-1035415-6.
  25. G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175--201. MR1182165. Zbl 0776.17012. DOI: https://doi.org/10.1143/PTP.102.175.
  26. G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141--182. MR1623674. Zbl 0915.17008. DOI: https://doi.org/10.1006/aima.1998.1729.
  27. G. Lusztig, Parabolic character sheaves. I, Mosc. Math. J. 4 (2004), no. 1, 153--179, 311. MR2074987. Zbl 1102.20030. DOI: https://doi.org/10.17323/1609-4514-2004-4-1-153-179. DOI: https://doi.org/10.17323/1609-4514-2004-4-1-153-179.
  28. G. Lusztig, Parabolic character sheaves. II, Mosc. Math. J. 4 (2004), no. 4, 869--896, 981. MR2124170. Zbl 1103.20041. DOI: https://doi.org/10.17323/1609-4514-2004-4-4-869-896.
  29. G. Lusztig, Quiver varieties and Weyl group actions, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 461--489. MR1775358. Zbl 0958.20036. DOI: https://doi.org/10.5802/aif.1762.
  30. G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365--421. MR1088333. Zbl 0738.17011. DOI: https://doi.org/10.1090/S0894-0347-1991-1088333-2.
  31. G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), no. 2, 129--139. MR1758244. Zbl 0983.17009. DOI: https://doi.org/ 10.1006/aima.1999.1873.
  32. D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer-Verlag, Berlin, 1994. xiv+292 pp. MR1304906. Zbl 0797.14004. DOI: https://link.springer. com/book/9783540569633.
  33. H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365--416. MR1302318. Zbl 0826.17026. DOI: https://doi.org/10.1215/S0012-7094-94-07613-8.
  34. H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, Amer. Math. Soc., Providence, RI, 1999. xii+132 pp. MR1711344. Zbl 0949.14001. DOI: https://doi.org/10.1090/ulect/018.
  35. H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515--560. MR1604167. Zbl 0970.17017. DOI: https://doi.org/10.1215/S0012-7094-98-09120-7.
  36. T. A. Nevins, Stability and Hamiltonian reduction for Grothendieck--Springer resolutions, University of Illinois at Urbana-Champaign, 2011, 1--25. rlhttp://www.math.uiuc.edu/~nevins/papers/b-hamiltonian-reduction-2011-0316.pdf.
  37. P. E. Newstead, Geometric invariant theory, in Moduli spaces and vector bundles, London Math. Soc. Lecture Note Ser. 359 (2009), 99--127. MR2537067. Zbl 1187.14054. DOI: https://doi.org/10.1017/CBO9781139107037.005.
  38. C. Procesi, The invariant theory of n × n matrices, Adv. Math. 19 (1976), no. 3, 306--381. MR0419491. Zbl 0331.15021. DOI: https://doi.org/10.1016/0001-8708(76)90027-X.
  39. R. Rouquier, 2-Kac-Moody algebras, 2008, 1--66. rlhttp://arxiv.org/pdf/0812.5023v1.
  40. A. Schofield, M. van den Bergh, Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.) 12 (2001), no. 1, 125--138. MR1908144. Zbl 1004.16012. DOI: https://doi.org/10.1016/S0019-3577(01)80010-0.






Mee Seong Im, ORCID: 0000-0003-1587-9145
Current address: Department of Mathematics, Johns Hopkins University,
Baltimore, MD 21218, USA.
Department of Mathematics, United States Naval Academy,
Annapolis, MD 21402, USA.
Department of Mathematical Sciences, United States Military Academy,
West Point, NY 10996, USA.
e-mail: meeseong@jhu.edu
https://sites.google.com/site/meeseongim/


Received: August 12, 2023; Accepted: March 14, 2026
Published electronically: March 18, 2026