Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T03:25:56.095Z Has data issue: false hasContentIssue false
  • English
  • Français

Langlands duality and global Springer theory

Part of: Linear algebraic groups and related topics Curves Abelian varieties and schemes (Co)homology theory

Published online by Cambridge University Press:  19 March 2012

Zhiwei Yun
Zhiwei Yun*
Affiliation:
MIT Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compare the cohomology of (parabolic) Hitchin fibers for Langlands dual groups G and G∨. The comparison theorem fits in the framework of the global Springer theory developed by the author. We prove that the stable parts of the parabolic Hitchin complexes for Langlands dual group are naturally isomorphic after passing to the associated graded of the perverse filtration. Moreover, this isomorphism intertwines the global Springer action on one hand and Chern class action on the other. Our result is inspired by the mirror symmetric viewpoint of geometric Langlands duality. Compared to the pioneer work in this subject by T. Hausel and M. Thaddeus, R. Donagi and T. Pantev, and N. Hitchin, our result is valid for more general singular fibers. The proof relies on a variant of Ngô’s support theorem, which is a key point in the proof of the Fundamental Lemma.

Keywords

Hitchin fibrationLanglands dualityT-duality

MSC classification

Secondary: 14H60: Vector bundles on curves and their moduli 20G35: Linear algebraic groups over adèles and other rings and schemes 14F20: Étale and other Grothendieck topologies and (co)homologies 14K30: Picard schemes, higher Jacobians
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 835 - 867
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Ari02]Arinkin, D., Fourier transform for quantized completely integrable systems, PhD thesis, Harvard University (2002).Google Scholar
[BBD82]Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque 100 (1982), 5–171.Google Scholar
[BB07]Braverman, A. and Bezrukavnikov, R., Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case, Pure Appl. Math. Q. 3 (2007), 153–179.Google Scholar
[dHM]de Cataldo, M. A., Hausel, T. and Migliorini, L., Topology of Hitchin systems and Hodge theory of character varieties: the case A 1, Ann. of Math. (2), to appear, math.AG/arXiv:1004.1420v2.Google Scholar
[DG02]Donagi, R. and Gaitsgory, D., The gerbe of Higgs bundles, Transform. Groups 7 (2002), 109–153.CrossRefGoogle Scholar
[DP06]Donagi, R. and Pantev, T., Langlands duality for Hitchin systems, Preprint (2006), arXiv:math.AG/0604617v2.Google Scholar
[HT03]Hausel, T. and Thaddeus, M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197–229.CrossRefGoogle Scholar
[Hit87]Hitchin, N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.CrossRefGoogle Scholar
[Hit07]Hitchin, N., Langlands duality and G 2 spectral curves, Q. J. Math. 58 (2007), 319–344.CrossRefGoogle Scholar
[KW07]Kapustin, A. and Witten, E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1–236.CrossRefGoogle Scholar
[Kle68]Kleiman, S., Algebraic cycles and Weil conjectures, in Dix exposés sur la Cohomologie des Schémas (North-Holland, Amsterdam, 1968).Google Scholar
[LN08]Laumon, G. and Ngô, B.-C., Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2) 168 (2008), 477–573.CrossRefGoogle Scholar
[Mum74]Mumford, D., Abelian varieties, Studies in Mathematics (Tata Institute of Fundamental Research, Bombay), vol. 5, second edition (Oxford University Press, London, 1974).Google Scholar
[Ngo06]Ngô, B.-C., Fibration de Hitchin et endoscopie, Invent. Math. 164 (2006), 399–453.CrossRefGoogle Scholar
[Ngo10]Ngô, B.-C., Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169.CrossRefGoogle Scholar
[Ngo11]Ngô, B.-C., Decomposition theorem and abelian fibration, in On the Stabilization of the Trace Formula, Stabilization of the trace formula, Shimura varieties, and arithmetic applications, vol. 1 (International Press, Somerville, MA, 2011), 253–264;http://fa.institut.math.jussieu.fr/node/44.Google Scholar
[SYZ96]Strominger, A., Yau, S.-T. and Zaslow, E., Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 243–259.CrossRefGoogle Scholar
[Yun09]Yun, Z., Towards a global springer theory III: endoscopy and Langlands duality, Preprint (2009), math.AG/arXiv:0904.3372.Google Scholar
[Yun11]Yun, Z., Global springer theory, Adv. Math. 228 (2011), 266–328.CrossRefGoogle Scholar