  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.
