Geometric braid group action on derived categories of coherent sheaves

Riche, Simon

doi:10.1090/S1088-4165-08-00325-7

Geometric braid group action on derived categories of coherent sheaves
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by Simon Riche; \\ with a joint appendix with Roman Bezrukavnikov

Represent. Theory 12 (2008), 131-169

DOI: https://doi.org/10.1090/S1088-4165-08-00325-7

Published electronically: March 10, 2008

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Abstract:

In this paper we give, for semi-simple groups without factors of type $\mathbf {G}_2$, a geometric construction of a braid group action on $\mathcal {D}^b \operatorname {Coh}(\widetilde {\mathfrak {g}})$ extending the action constructed by Bezrukavnikov, Mirković and Rumynin in the context of localization in positive characteristic. It follows that this action extends to characteristic zero, where it also has some nice representation-theoretic interpretations. The argument uses a presentation of the affine braid group analogous to the “Bernstein presentation” of the corresponding Hecke algebra (this presentation was suggested by Lusztig; it is worked out in the appendix, written jointly with Roman Bezrukavnikov).

References

Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
R. Bezrukavnikov, Perverse coherent sheaves (after Deligne), preprint on arXiv: AG/0005152v1.
Roman Bezrukavnikov, Noncommutative counterparts of the Springer resolution, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1119–1144. MR 2275638
R. Bezrukavnikov, I. Mirković, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, with an appendix by R. Bezrukavnikov and S. Riche, preprint on arXiv: RT/0205144v8, to appear in Ann. of Math.
Roman Bezrukavnikov, Ivan Mirković, and Dmitriy Rumynin, Singular localization and intertwining functors for reductive Lie algebras in prime characteristic, Nagoya Math. J. 184 (2006), 1–55. MR 2285230, DOI 10.1017/S0027763000009302
A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2107324, DOI 10.1007/b137486
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
A. Căldăraru, The Mukai pairing, I: the Hochschild structure, preprint on arXiv: AG/0308079v2.
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
V. Ginsburg, ${\mathfrak {G}}$-modules, Springer’s representations and bivariant Chern classes, Adv. in Math. 61 (1986), no. 1, 1–48. MR 847727, DOI 10.1016/0001-8708(86)90064-2
V. Ginsburg, “Lagrangian” construction for representations of Hecke algebras, Adv. in Math. 63 (1987), no. 1, 100–112. MR 871082, DOI 10.1016/0001-8708(87)90064-8
A. Grothendieck, EGA I, avec la collaboration de J. Dieudonné, Springer, 1971.
A. Grothendieck, EGA III$_1$, Étude cohomologique des faisceaux cohérents, avec la collaboration de J. Dieudonné, Publ. Math. Inst. Hautes Études Sci. 11 (1961).
S. Gukov, E. Witten, Gauge theory, ramification, and the geometric Langlands program, preprint on arXiv: hep-th/0612073v1.
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016, DOI 10.1007/BF02684396
M. Khovanov, R. Thomas, Braid cobordisms, triangulated categories, and flag varieties, preprint on arXiv: QA/0609335v1.
George Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Adv. in Math. 37 (1980), no. 2, 121–164. MR 591724, DOI 10.1016/0001-8708(80)90031-6
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
G. Lusztig, Bases in equivariant $K$-theory, Represent. Theory 2 (1998), 298–369. MR 1637973, DOI 10.1090/S1088-4165-98-00054-5
G. Lusztig, Bases in equivariant $K$-theory. II, Represent. Theory 3 (1999), 281–353. MR 1714628, DOI 10.1090/S1088-4165-99-00083-7
George Lusztig, Notes on affine Hecke algebras, Iwahori-Hecke algebras and their representation theory (Martina-Franca, 1999) Lecture Notes in Math., vol. 1804, Springer, Berlin, 2002, pp. 71–103. MR 1979925, DOI 10.1007/978-3-540-36205-0_{3}
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. MR 1976581, DOI 10.1017/CBO9780511542824
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Ivan Mirković and Dmitriy Rumynin, Centers of reduced enveloping algebras, Math. Z. 231 (1999), no. 1, 123–132. MR 1696760, DOI 10.1007/PL00004719
S. Riche, Koszul duality and modular representations of semi-simple Lie algebras, in preparation.
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4

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Representation Theory

Geometric braid group action on derived categories of coherent sheaves
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Abstract: