Springer theory via the Hitchin fibration

TL;DR

This paper connects Springer theory of Weyl group representations with symplectic topology by describing Lagrangian branes in cotangent bundles, offering insights into the geometric Langlands program.

Contribution

It translates Springer theory into symplectic topology using Fukaya categories and Fourier transforms, providing a novel perspective on the geometric Langlands correspondence.

Findings

Lagrangian branes produce Springer perverse sheaves

Fourier transform analysis in Fukaya categories

Toy model for Hitchin fiber quantization

Abstract

In this paper, we translate the Springer theory of Weyl group representations into 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.