| In this dissertation, we give a description of the equivariant derived category of D-modules on a complex reductive group G in terms of cuspidal local systems on Levi subgroups of G. The main tools we use are parabolic induction (or Eisenstein series) and parabolic restriction (or constant term) functors, together with a version of the Mackey formula. We show that parabolic induction and restriction are bi-adjoint, and produce an orthogonal decomposition of the derived category into blocks generated by inductions of cuspidal D-modules from Levi subgroups. The composition of induction and restriction defines a monad on the category of equivariant cuspidal D-modules on each Levi, L which we identify with the action of the relative Weyl group W(L). It follows from the Barr-Beck-Lurie theorem that the block containing inductions from L is equivalent to cuspidal D-modules on L together with an action of W(L).;These results can be considered as an extension of Lusztig's generalized Springer correspondence. They are closely related to the derived Springer correspondence of Rider (a version of which can be obtained by restricting to D-modules supported on the unipotent cone). The adjoint quotient stack G/adG is an open substack of Bun G(C), where C is a nodal curve of genus 1. In this way, the results can be considered as a step towards understanding the geometric Langlands correspondence for genus 1 curves. |
