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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: zyun@math.mit.edu)
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Abstract

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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

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