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Singular Localization and Intertwining Functors for Reductive Lie Algebras in Prime Characteristic

Published online by Cambridge University Press:  11 January 2016

Roman Bezrukavnikov ,
Ivan Mirković  and
Dmitriy Rumynin
Roman Bezrukavnikov
Affiliation:
Department of MathematicsMassachusetts Institute of Technology77 Massachusetts ave. Cambridge, MA 02139U.S.A.bezrukav@math.mit.edu
Ivan Mirković
Affiliation:
Department of Mathematics and StatisticsUniversity of MassachusettsAmherst, MA 01003U.S.A.mirkovic@math.umass.edu
Dmitriy Rumynin
Affiliation:
Mathematics DepartmentUniversity of WarwickCoventry, CV4 7ALEnglandrumynin@maths.warwick.ac.uk
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Abstract

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In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.

The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

Keywords

17B5016E2018F99
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 183 , 2006 , pp. 1 - 55
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[BB1] Beilinson, A. and Bernstein, A., Localisation de g-modules, C. R. Acad. Sci. Paris Séer. I Math., 292 (1981), 15–18.Google Scholar
[BB2] Beilinson, A. and Bernstein, A., A generalization of Casselman’s submodule theorem, Representation theory of reductive groups (Park City, Utah, 1982), Progr. Math. 40, Birkhäuser, Boston (1983), pp. 35–52.Google Scholar
[BeG] Beilinson, A. and Ginzburg, V., Wall-crossing functors and D-modules, Represent. Theory, 3 (1999), 1–31.CrossRefGoogle Scholar
[B] Bezrukavnikov, R., Noncommutative counterparts of the Springer resolution, Proceeding of the International Congress of Mathematicians, Madrid, Spain, 2006, vol. 2, pp. 1119–1144.Google Scholar
[BrBr] Bezrukavnikov, R. and Braverman, A., Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case, Quartely J. Pure Appl. Math., to appear.Google Scholar
[BMR] Bezrukavnikov, R., MirkoviĆ, I. and Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic, with an appendix by S. Riche, preprint math.RT/0205144, to appear in Ann. Math.Google Scholar
[BK] Bondal, A. and Kapranov, M., Representable functors, Serre functors, and mutations, Izv. Ak. Nauk, 53 (1989), 1183–1205; translation in Math. USSR-Izv., 35 (1990), 519–541.Google Scholar
[Bo] Borel, A. et. al., Algebraic D-modules, Perspectives in Mathematics, 2, Academic Press, Boston, 1987.Google Scholar
[BKR] Bridgeland, T., King, A. and Reid, M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc., 14 (2001), no. 3, 535–554 (preprint version Mukai implies McKay available as math.AG/9908027 at xxx.lanl.gov).CrossRefGoogle Scholar
[BrKu] Brion, M. and Kumar, S., Frobenius splitting methods in geometry and representation theory, Progress Math. 231, Birkhäuser, Boston, 2005.Google Scholar
[Bro] Broer, B., Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety, Lie Theory and Geometry, Progress Math. 123, Birkhäuser, Boston (1994), pp. 1–19.Google Scholar
[BrGo] Brown, K. and Goodearl, K., Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension, J. Algebra, 198 (1997), 240–265.CrossRefGoogle Scholar
[BG] Brown, K. and Gordon, I., The ramification of centres: Lie algebras in positive characteristic and quantized enveloping algebras, Math. Z., 238 (2001), 733–779.CrossRefGoogle Scholar
[CG] Chriss, N. and Ginzburg, V., Representation theory and complex geometry, Birkhäuser, Boston, 1997.Google Scholar
[De] Demazure, M., Invariants syméetriques entiers des groupes de Weyl et torsion, Invent. Math., 21 (1973), 287–301.CrossRefGoogle Scholar
[HKR] Hashimoto, Y., Kaneda, M. and Rumynin, D., On localization of D-modules, Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, Contemp. Math. 413, AMS, Providence (2006), pp. 43–62.Google Scholar
[Ja] Jantzen, J., Representations of Lie algebras in prime characteristic, Representation theories and algebraic geometry, Proceedings NATO ASI (Montreal, 1997), Kluwer, Dordrecht (1998), pp. 185–235.Google Scholar
[KW] Kac, V. and Weisfeiler, B., Coadjoint action of a semisimple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math., 38 (1976), 136–151.CrossRefGoogle Scholar
[Lu] Lusztig, G., Bases in equivariant K-theory, Represent. Theory, 2 (1998), 298–369. Bases in equivariant K-theory II, Represent. Theory, 3 (1999), 281–353.CrossRefGoogle Scholar
[Lu1] Lusztig, G., Hecke algebras and Jantzen’s generic decomposition patterns, Adv. in Math., 37 (1980), 121–164.CrossRefGoogle Scholar
[Mi] MiliˇciĆ, D., Localization and Representation Theory of Reductive Lie Groups, available at http://www.math.utah.edu/~milicic.Google Scholar
[Mln] Milne, J. S., Etale cohomology, Princeton Math. Series 33, Princeton U. Press, 1980.Google Scholar
[MR] MirkoviĆ, I. and Rumynin, D., Centers of reduced enveloping algebras, Math. Z., 231 (1999), 123–132.CrossRefGoogle Scholar
[OV] Ogus, A. and Vologodsky, V., Nonabelian Hodge theory in characteristic p, preprint math.AG/0507476.Google Scholar
[R] Reid, M., McKay correspondence, preprint alg-geom/9702016.Google Scholar
[Sk] Skorobogatov, A., Torsors and rational points, Cambridge U. Press, Cambridge, 2001.Google Scholar
[Ve] Veldkamp, F., The center of the universal enveloping algebra of a Lie algebra in é characteristic p, Ann. Sci. Ecole Norm. Sup. (4), 5 (1972), 217–240.Google Scholar