Springer theory via the Hitchin fibration

Abstract

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 [formula omitted] 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. © 2011, Foundation Compositio Mathematica. All rights reserved.