Affine braid group actions on derived categories of Springer resolutions

Bezrukavnikov, Roman; Riche, Simon

Affine braid group actions on derived categories of Springer resolutions
[Actions du groupe de tresses affine sur les catégories dérivées des résolutions de Springer]

Bezrukavnikov, Roman ; Riche, Simon

Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012), p. 535-599 / Harvested from Numdam

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Abstract

Dans cet article nous construisons et étudions une action du groupe de tresses affine associé à un groupe algébrique semi-simple sur les catégories dérivées de faisceaux cohérents sur diverses variétés liées à la résolution de Springer du cône nilpotent. En particulier, nous décrivons explicitement l’action du groupe de tresses d’Artin. Cette action est une « version catégorique » de la construction géométrique de l’algèbre de Hecke affine due à Kazhdan-Lusztig et Ginzburg, et est utilisée par le premier auteur et I. Mirković au cours de la preuve des conjectures de Lusztig sur la $K$ -théorie équivariante des fibres de Springer.

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.

Publié le : 2012-01-01

MR 3059241

DOI : https://doi.org/10.24033/asens.2173
Classification: 20G99, 14M15, 17B20, 18E30, 20F36
Mots clés: groupe de tresses, groupe algébrique réductif, algèbre de Lie, résolution de Springer, algèbre de Hecke affine, dg-schéma, transformée de Fourier-Mukai

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     author = {Bezrukavnikov, Roman and Riche, Simon},
     title = {Affine braid group actions on derived categories of Springer resolutions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {45},
     year = {2012},
     pages = {535-599},
     doi = {10.24033/asens.2173},
     mrnumber = {3059241},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2012_4_45_4_535_0}
}

Bezrukavnikov, Roman; Riche, Simon. Affine braid group actions on derived categories of Springer resolutions. Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012) pp. 535-599. doi : 10.24033/asens.2173. http://gdmltest.u-ga.fr/item/ASENS_2012_4_45_4_535_0/

Bibliographie

[1] L. Alonso Tarrío, A. Jeremías López & J. Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. 30 (1997), 1-39. | Numdam | MR 1422312 | Zbl 0894.14002

[2] R. Anno, Spherical functors, preprint arXiv:0711.4409.

[3] D. Arinkin & R. Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J. 10 (2010), 3-29. | MR 2668828 | Zbl 1205.18010

[4] M. Artin & M. Van Den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), 249-271. | MR 1067406 | Zbl 0717.14001

[5] A. Beilinson, R. Bezrukavnikov & I. Mirković, Tilting exercises, Mosc. Math. J. 4 (2004), 547-557. | MR 2119139 | Zbl 1075.14015

[6] J. Bernstein & V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer, 1994. | MR 1299527 | Zbl 0808.14038

[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. | MR 354655 | Zbl 0218.14001

[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. | MR 2249802 | Zbl 1123.17002

[9] R. Bezrukavnikov, Noncommutative counterparts of the Springer resolution, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, 1119-1144. | MR 2275638 | Zbl 1135.17011

[10] R. Bezrukavnikov, On two realizations of an affine Hecke algebra, preprint arXiv:1209.0403.

[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. | MR 3092472 | Zbl pre06220725

[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. | MR 2285230 | Zbl 1125.17006

[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. | MR 2415389 | Zbl 1220.17009

[14] R. Bezrukavnikov & S. Riche, Presentation of $B_{aff}^{'}$ , appendix to [54].

[15] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6, Hermann, 1968 ; réédition Springer, 2006. | Zbl 1181.17001

[16] C. Brav & H. Thomas, Braid groups and Kleinian singularities, Math. Ann. 351 (2011), 1005-1017. | MR 2854121 | Zbl 1264.14026

[17] T. Bridgeland, Stability conditions and Kleinian singularities, Int. Math. Res. Not. 2009 (2009), 4142-4157. | MR 2549952 | Zbl 1228.14012

[18] M. Brion, Multiplicity-free subvarieties of flag varieties, in Commutative algebra (Grenoble/Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc., 2003, 13-23. | MR 2011763 | Zbl 1052.14055

[19] M. Brion & S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Math. 231, Birkhäuser, 2005. | MR 2107324 | Zbl 1072.14066

[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. | MR 1872572 | Zbl 1037.17011

[21] S. Cautis & J. Kamnitzer, Braiding via geometric Lie algebra actions, Compos. Math. 148 (2012), 464-506. | MR 2904194 | Zbl 1249.14005

[22] N. Chriss & V. Ginzburg, Representation theory and complex geometry, Birkhäuser, 1997. | MR 1433132 | Zbl 0879.22001

[23] I. Ciocan-Fontanine & M. Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup. 34 (2001), 403-440. | Numdam | MR 1839580 | Zbl 1050.14042

[24] P. Deligne, Action du groupe des tresses sur une catégorie, Invent. Math. 128 (1997), 159-175. | MR 1437497 | Zbl 0879.57017

[25] C. Dodd, Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras, preprint arXiv:1108.4028. | MR 2982382

[26] D. Eisenbud, Commutative algebra. With a view towards algebraic geometry, Graduate Texts in Math. 150, Springer, 1995. | MR 1322960 | Zbl 0819.13001

[27] W. Fulton, Intersection theory, second éd., Ergebn. Math. Grenzg. 2, Springer, 1998. | MR 1644323 | Zbl 0885.14002

[28] V. Ginsburg, $𝔊$ -modules, Springer's representations and bivariant Chern classes, Adv. in Math. 61 (1986), 1-48. | MR 847727 | Zbl 0601.22008

[29] V. Ginzburg, Variations on themes of Kostant, Transform. Groups 13 (2008), 557-573. | MR 2452606 | Zbl 1169.17002

[30] V. Ginzburg, Harish-Chandra bimodules for quantized Slodowy slices, Represent. Theory 13 (2009), 236-271. | MR 2515934 | Zbl 1250.17007

[31] M. Grinberg, A generalization of Springer theory using nearby cycles, Represent. Theory 2 (1998), 410-431. | MR 1657203 | Zbl 0938.22011

[32] A. Grothendieck, Ega Iii, Étude cohomologique des faisceaux cohérents (première partie), Publ. Math. IHÉS 11 (1961). | Numdam | Zbl 0122.16102

[33] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20, Springer, 1966. | MR 222093 | Zbl 0212.26101

[34] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977. | MR 463157 | Zbl 0367.14001

[35] J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs 43, Amer. Math. Soc., 1995. | MR 1343976 | Zbl 0834.20048

[36] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 2006. | MR 2244106 | Zbl 1095.14002

[37] A. Ishii, K. Ueda & H. Uehara, Stability conditions on $A_{n}$ -singularities, J. Differential Geom. 84 (2010), 87-126. | MR 2629510 | Zbl 1198.14020

[38] J. C. Jantzen, Subregular nilpotent representations of Lie algebras in prime characteristic, Represent. Theory 3 (1999), 153-222. | MR 1703320 | Zbl 0998.17003

[39] J. C. Jantzen, Representations of algebraic groups, second éd., Mathematical Surveys and Monographs 107, Amer. Math. Soc., 2003. | MR 2015057 | Zbl 1034.20041

[40] J. C. Jantzen, Nilpotent orbits in representation theory, in Lie theory, Progr. Math. 228, Birkhäuser, 2004, 1-211. | MR 2042689 | Zbl 1169.14319

[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. | MR 417308 | Zbl 0324.17001

[42] M. Kashiwara & Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9-36. | MR 1458969 | Zbl 0901.17006

[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. | MR 751138 | Zbl 0611.22008

[44] B. Keller, Derived categories and their uses, in Handbook of algebra, Vol. 1, North-Holland, 1996, 671-701. | MR 1421815 | Zbl 0862.18001

[45] M. Khovanov & R. Thomas, Braid cobordisms, triangulated categories, and flag varieties, Homology, Homotopy Appl. 9 (2007), 19-94. | MR 2366943 | Zbl 1119.18008

[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. | MR 2490557 | Zbl 1163.14001

[47] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169-178. | MR 641425 | Zbl 0473.20029

[48] G. Lusztig, Bases in equivariant $K$ -theory, Represent. Theory 2 (1998), 298-369. | MR 1637973 | Zbl 0901.20034

[49] G. Lusztig, Bases in equivariant $K$ -theory. II, Represent. Theory 3 (1999), 281-353. | MR 1714628 | Zbl 0999.20036

[50] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157, Cambridge Univ. Press, 2003. | MR 1976581 | Zbl 1024.33001

[51] H. Matsumura, Commutative algebra, second éd., Mathematics Lecture Note Series 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. | MR 575344 | Zbl 0441.13001

[52] I. Mirković & S. Riche, Linear Koszul duality and affine Hecke algebras, preprint arXiv:0903.0678. | MR 2581249 | Zbl 1275.14014

[53] I. Mirković & D. Rumynin, Centers of reduced enveloping algebras, Math. Z. 231 (1999), 123-132. | MR 1696760 | Zbl 0932.17020

[54] S. Riche, Geometric braid group action on derived categories of coherent sheaves, Represent. Theory 12 (2008), 131-169. | MR 2390670 | Zbl 1156.14014

[55] S. Riche, Koszul duality and modular representations of semisimple Lie algebras, Duke Math. J. 154 (2010), 31-134. | MR 2668554 | Zbl 1264.17005

[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. | MR 2258045 | Zbl 1162.20301

[57] P. Seidel & R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37-108. | MR 1831820 | Zbl 1092.14025

[58] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math. 815, Springer, 1980. | MR 584445 | Zbl 0441.14002

[59] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121-154. | Numdam | MR 932640 | Zbl 0636.18006

[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. | Zbl 1242.20007

[61] T. Xue, Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence, Represent. Theory 13 (2009), 609-635. | MR 2558787 | Zbl 1250.17009

[62] T. Xue, Nilpotent orbits in bad characteristic and the Springer correspondence, Thèse, Massachusetts Institute of Technology, 2010. | MR 2814044