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Springer theory via the Hitchin fibration

Part of:Lie groupsSymplectic geometry, contact geometrySingularities

Published online by Cambridge University Press:  29 July 2011

David Nadler
David Nadler*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA (email: nadler@math.northwestern.edu)
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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.

Keywords

Fukaya categoryFourier transformHitchin fibration

MSC classification

Secondary:53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category32S60: Stratifications; constructible sheaves; intersection cohomology22E57: Geometric Langlands program

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011, pp. 1635 - 1670
Copyright
Copyright © Foundation Compositio Mathematica 2011

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