Affine braid group actions on derived categories of Springer resolutions

Roman Bezrukavnikov; Simon Riche

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 4, page 535-599
  • ISSN: 0012-9593

Abstract

top
In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course of the proof of Lusztig’s conjectures on equivariant K -theory of Springer fibers.

How to cite

top

Bezrukavnikov, Roman, and Riche, Simon. "Affine braid group actions on derived categories of Springer resolutions." Annales scientifiques de l'École Normale Supérieure 45.4 (2012): 535-599. <http://eudml.org/doc/272154>.

@article{Bezrukavnikov2012,
abstract = {In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course of the proof of Lusztig’s conjectures on equivariant $K$-theory of Springer fibers.},
author = {Bezrukavnikov, Roman, Riche, Simon},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {braid group; reductive algebraic group; Lie algebra; Springer resolution; affine Hecke algebra; dg-scheme; Fourier-Mukai transform},
language = {eng},
number = {4},
pages = {535-599},
publisher = {Société mathématique de France},
title = {Affine braid group actions on derived categories of Springer resolutions},
url = {http://eudml.org/doc/272154},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Bezrukavnikov, Roman
AU - Riche, Simon
TI - Affine braid group actions on derived categories of Springer resolutions
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 4
SP - 535
EP - 599
AB - In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course of the proof of Lusztig’s conjectures on equivariant $K$-theory of Springer fibers.
LA - eng
KW - braid group; reductive algebraic group; Lie algebra; Springer resolution; affine Hecke algebra; dg-scheme; Fourier-Mukai transform
UR - http://eudml.org/doc/272154
ER -

References

top
  1. [1] L. Alonso Tarrío, A. Jeremías López & J. Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup.30 (1997), 1–39. Zbl0894.14002MR1422312
  2. [2] R. Anno, Spherical functors, preprint arXiv:0711.4409. Zbl06506323
  3. [3] D. Arinkin & R. Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J.10 (2010), 3–29. Zbl1205.18010MR2668828
  4. [4] M. Artin & M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra133 (1990), 249–271. Zbl0717.14001MR1067406
  5. [5] A. Beilinson, R. Bezrukavnikov & I. Mirković, Tilting exercises, Mosc. Math. J.4 (2004), 547–557. Zbl1075.14015MR2119139
  6. [6] J. Bernstein & V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer, 1994. Zbl0808.14038MR1299527
  7. [7] P. Berthelot, A. Grothendieck & L. Illusie, Théorie des intersections et théorème de Riemann-Roch, in Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Math. 225, Springer, 1971. Zbl0218.14001MR354655
  8. [8] R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and t -structures on derived categories of coherent sheaves, Invent. Math.166 (2006), 327–357. Zbl1123.17002MR2249802
  9. [9] R. Bezrukavnikov, Noncommutative counterparts of the Springer resolution, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, 1119–1144. Zbl1135.17011MR2275638
  10. [10] R. Bezrukavnikov, On two realizations of an affine Hecke algebra, preprint arXiv:1209.0403. 
  11. [11] R. Bezrukavnikov & I. Mirković, Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution, preprint arXiv:1001.2562, to appear in Ann. of Math. Zbl1293.17021MR3092472
  12. [12] R. Bezrukavnikov, I. Mirković & D. Rumynin, Singular localization and intertwining functors for reductive Lie algebras in prime characteristic, Nagoya Math. J.184 (2006), 1–55. Zbl1125.17006MR2285230
  13. [13] R. Bezrukavnikov, I. Mirković & D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math.167 (2008), 945–991. Zbl1220.17009MR2415389
  14. [14] R. Bezrukavnikov & S. Riche, Presentation of B aff ' , appendix to [54]. 
  15. [15] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6, Hermann, 1968 ; réédition Springer, 2006. Zbl1181.17001
  16. [16] C. Brav & H. Thomas, Braid groups and Kleinian singularities, Math. Ann.351 (2011), 1005–1017. Zbl1264.14026MR2854121
  17. [17] T. Bridgeland, Stability conditions and Kleinian singularities, Int. Math. Res. Not.2009 (2009), 4142–4157. Zbl1228.14012MR2549952
  18. [18] M. Brion, Multiplicity-free subvarieties of flag varieties, in Commutative algebra (Grenoble/Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc., 2003, 13–23. Zbl1052.14055MR2011763
  19. [19] M. Brion & S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Math. 231, Birkhäuser, 2005. Zbl1072.14066MR2107324
  20. [20] K. A. Brown & I. Gordon, The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z.238 (2001), 733–779. Zbl1037.17011MR1872572
  21. [21] S. Cautis & J. Kamnitzer, Braiding via geometric Lie algebra actions, Compos. Math.148 (2012), 464–506. Zbl1249.14005MR2904194
  22. [22] N. Chriss & V. Ginzburg, Representation theory and complex geometry, Birkhäuser, 1997. Zbl0879.22001MR1433132
  23. [23] I. Ciocan-Fontanine & M. Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup.34 (2001), 403–440. Zbl1050.14042MR1839580
  24. [24] P. Deligne, Action du groupe des tresses sur une catégorie, Invent. Math.128 (1997), 159–175. Zbl0879.57017MR1437497
  25. [25] C. Dodd, Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras, preprint arXiv:1108.4028. MR2982382
  26. [26] D. Eisenbud, Commutative algebra. With a view towards algebraic geometry, Graduate Texts in Math. 150, Springer, 1995. Zbl0819.13001MR1322960
  27. [27] W. Fulton, Intersection theory, second éd., Ergebn. Math. Grenzg. 2, Springer, 1998. Zbl0885.14002MR1644323
  28. [28] V. Ginsburg, 𝔊 -modules, Springer’s representations and bivariant Chern classes, Adv. in Math. 61 (1986), 1–48. Zbl0601.22008MR847727
  29. [29] V. Ginzburg, Variations on themes of Kostant, Transform. Groups13 (2008), 557–573. Zbl1169.17002MR2452606
  30. [30] V. Ginzburg, Harish-Chandra bimodules for quantized Slodowy slices, Represent. Theory13 (2009), 236–271. Zbl1250.17007MR2515934
  31. [31] M. Grinberg, A generalization of Springer theory using nearby cycles, Represent. Theory2 (1998), 410–431. Zbl0938.22011MR1657203
  32. [32] A. Grothendieck, EGA III, Étude cohomologique des faisceaux cohérents (première partie), Publ. Math. IHÉS 11 (1961). Zbl0122.16102
  33. [33] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20, Springer, 1966. Zbl0212.26101MR222093
  34. [34] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977. Zbl0367.14001MR463157
  35. [35] J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs 43, Amer. Math. Soc., 1995. Zbl0834.20048MR1343976
  36. [36] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 2006. Zbl1095.14002MR2244106
  37. [37] A. Ishii, K. Ueda & H. Uehara, Stability conditions on A n -singularities, J. Differential Geom.84 (2010), 87–126. Zbl1198.14020MR2629510
  38. [38] J. C. Jantzen, Subregular nilpotent representations of Lie algebras in prime characteristic, Represent. Theory3 (1999), 153–222. Zbl0998.17003MR1703320
  39. [39] J. C. Jantzen, Representations of algebraic groups, second éd., Mathematical Surveys and Monographs 107, Amer. Math. Soc., 2003. Zbl1034.20041MR2015057
  40. [40] J. C. Jantzen, Nilpotent orbits in representation theory, in Lie theory, Progr. Math. 228, Birkhäuser, 2004, 1–211. Zbl1169.14319MR2042689
  41. [41] V. Kac & B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p , Indag. Math.38 (1976), 136–151. Zbl0324.17001MR417308
  42. [42] M. Kashiwara & Y. Saito, Geometric construction of crystal bases, Duke Math. J.89 (1997), 9–36. Zbl0901.17006MR1458969
  43. [43] M. Kashiwara & T. Tanisaki, The characteristic cycles of holonomic systems on a flag manifold related to the Weyl group algebra, Invent. Math.77 (1984), 185–198. Zbl0611.22008MR751138
  44. [44] B. Keller, Derived categories and their uses, in Handbook of algebra, Vol. 1, North-Holland, 1996, 671–701. Zbl0862.18001MR1421815
  45. [45] M. Khovanov & R. Thomas, Braid cobordisms, triangulated categories, and flag varieties, Homology, Homotopy Appl. 9 (2007), 19–94. Zbl1119.18008MR2366943
  46. [46] J. Lipman, Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Math. 1960, Springer, 2009, 1–259. Zbl1163.14001MR2490557
  47. [47] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math.42 (1981), 169–178. Zbl0473.20029MR641425
  48. [48] G. Lusztig, Bases in equivariant K -theory, Represent. Theory2 (1998), 298–369. Zbl0901.20034MR1637973
  49. [49] G. Lusztig, Bases in equivariant K -theory. II, Represent. Theory 3 (1999), 281–353. Zbl0999.20036MR1714628
  50. [50] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157, Cambridge Univ. Press, 2003. Zbl1024.33001MR1976581
  51. [51] H. Matsumura, Commutative algebra, second éd., Mathematics Lecture Note Series 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. Zbl0441.13001MR575344
  52. [52] I. Mirković & S. Riche, Linear Koszul duality and affine Hecke algebras, preprint arXiv:0903.0678. Zbl1275.14014MR2581249
  53. [53] I. Mirković & D. Rumynin, Centers of reduced enveloping algebras, Math. Z.231 (1999), 123–132. Zbl0932.17020MR1696760
  54. [54] S. Riche, Geometric braid group action on derived categories of coherent sheaves, Represent. Theory12 (2008), 131–169. Zbl1156.14014MR2390670
  55. [55] S. Riche, Koszul duality and modular representations of semisimple Lie algebras, Duke Math. J.154 (2010), 31–134. Zbl1264.17005MR2668554
  56. [56] R. Rouquier, Categorification of 𝔰𝔩 2 and braid groups, in Trends in representation theory of algebras and related topics, Contemp. Math. 406, Amer. Math. Soc., 2006, 137–167. Zbl1162.20301MR2258045
  57. [57] P. Seidel & R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J.108 (2001), 37–108. Zbl1092.14025MR1831820
  58. [58] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math. 815, Springer, 1980. Zbl0441.14002MR584445
  59. [59] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math.65 (1988), 121–154. Zbl0636.18006MR932640
  60. [60] M. Varagnolo & É. Vasserot, Double affine Hecke algebras and affine flag manifolds, I, in Affine flag manifolds and principal bundles (A. Schmidt, éd.), Birkhäuser, 2010. Zbl1242.20007
  61. [61] T. Xue, Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence, Represent. Theory13 (2009), 609–635. Zbl1250.17009MR2558787
  62. [62] T. Xue, Nilpotent orbits in bad characteristic and the Springer correspondence, Thèse, Massachusetts Institute of Technology, 2010. MR2814044

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.