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Springer theory via the Hitchin fibration

Published online by Cambridge University Press:  29 July 2011

David Nadler*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA (email: [email protected])
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Abstract

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We develop the Springer theory of Weyl group representations in the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient 𝔤/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the geometric Langlands program.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[ALP94]Audin, M., Lalonde, F. and Polterovich, L., Symplectic rigidity: Lagrangian submanifolds, in Holomorphic curves in symplectic geometry, Progress in Mathematics, vol. 117 (Birkhäuser, Basel, 1994), 271321.CrossRefGoogle Scholar
[BD]Beilinson, A. and Drinfeld, V., Quantization of Hitchin Hamiltonians and Hecke eigensheaves, Preprint.Google Scholar
[BM88]Bierstone, E. and Milman, P., Semianalytic and subanalytic sets, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 542.CrossRefGoogle Scholar
[BM81]Borho, W. and MacPherson, R., Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 707710.Google Scholar
[Dri04]Drinfeld, V., DG quotients of DG categories, J. Algebra 272 (2004), 643691.CrossRefGoogle Scholar
[FO97]Fukaya, K. and Oh, Y.-G., Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997), 96180.CrossRefGoogle Scholar
[FOOO09]Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K., Lagrangian intersection Floer theory: anomaly and obstruction, Part I, AMS/IP Studies in Advanced Mathematics, vol. 46 (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009).CrossRefGoogle Scholar
[Gin83]Ginzburg, V., Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 249252.Google Scholar
[Gri98]Grinberg, M., A generalization of Springer theory using nearby cycles, Represent. Theory 2 (1998), 410431 (electronic).CrossRefGoogle Scholar
[HK84]Hotta, R. and Kashiwara, M., The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), 327358.CrossRefGoogle Scholar
[KW07]Kapustin, A. and Witten, E., Electric–magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1236.CrossRefGoogle Scholar
[KS94]Kashiwara, M. and Schapira, P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292 (Springer, Berlin, 1994).Google Scholar
[KL80]Kazhdan, D. and Lusztig, G., A topological approach to Springer’s representations, Adv. Math. 38 (1980), 222228.CrossRefGoogle Scholar
[Kel06]Keller, B., On differential graded categories, International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), 151190.Google Scholar
[Lus81]Lusztig, G., Green polynomials and singularities of unipotent classes, Adv. Math. 42 (1981), 169178.CrossRefGoogle Scholar
[Nad09]Nadler, D., Microlocal branes are constructible sheaves, Selecta Math. 15 (2009), 563619.CrossRefGoogle Scholar
[NZ09]Nadler, D. and Zaslow, E., Constructible sheaves and the Fukaya category, J. Amer. Math. Soc. 22 (2009), 233286.CrossRefGoogle Scholar
[Oh]Oh, Y.-G., Unwrapped continuation invariance in Lagrangian Floer theory: energy and C 0estimates, arXiv:0910.1131.Google Scholar
[Sei08]Seidel, P., Fukaya categories and Picard–Lefschetz theory, Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2008).CrossRefGoogle Scholar
[Sik94]Sikorav, J.-C., Some properties of holomorphic curves in almost complex manifolds, in Holomorphic curves in symplectic geometry (Birkhäuser, Basel, 1994), 165189.CrossRefGoogle Scholar
[Spr76]Springer, T. A., Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173207.CrossRefGoogle Scholar
[Spr78]Springer, T. A., A construction of representations of Weyl groups, Invent. Math. 44 (1978), 279293.CrossRefGoogle Scholar
[Spr82]Springer, T. A., Quelques applications de la cohomologie d’intersection, in Bourbaki seminar, vol. 1981/1982, Asterique, vols. 92–93 (Soc. Math. France, Paris, 1982), 249273.Google Scholar
[Sta63]Stasheff, J. D., Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275312.Google Scholar
[vM96]van den Dries, L. and Miller, C., Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497539.CrossRefGoogle Scholar