Affine braid group actions on derived categories of Springer resolutions

Bezrukavnikov, Roman; Riche, Simon

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 4, pp. 535-599.

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.

Classification : 20G99, 14M15, 17B20, 18E30, 20F36
Keywords: braid group, reductive algebraic group, Lie algebra, Springer resolution, affine Hecke algebra, dg-scheme, Fourier-Mukai transform
Mot clés : groupe de tresses, groupe algébrique réductif, algèbre de Lie, résolution de Springer, algèbre de Hecke affine, dg-schéma, transformée de Fourier-Mukai

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

[56] R. Rouquier, Categorification of ${\mathrm{𝔰𝔩}}_2$ and braid groups, in Trends in representation theory of algebras and related topics, Contemp. Math. 406, Amer. Math. Soc., 2006, 137-167. | MR | Zbl

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

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

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

[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

[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 | Zbl

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