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Selecta Mathematica

, Volume 24, Issue 5, pp 4223–4277 | Cite as

Generalized Springer theory for D-modules on a reductive Lie algebra

  • Sam Gunningham
Article
  • 8 Downloads

Abstract

Given a reductive group G, we give a description of the abelian category of G-equivariant D-modules on \(\mathfrak {g}={{\mathrm{Lie}}}(G)\), which specializes to Lusztig’s generalized Springer correspondence upon restriction to the nilpotent cone. More precisely, the category has an orthogonal decomposition in to blocks indexed by cuspidal data \((L,\mathcal {E})\), consisting of a Levi subgroup L, and a cuspidal local system \(\mathcal {E}\) on a nilpotent L-orbit. Each block is equivalent to the category of D-modules on the center \(\mathfrak {z}(\mathfrak {l})\) of \(\mathfrak {l}\) which are equivariant for the action of the relative Weyl group \(N_G(L)/L\). The proof involves developing a theory of parabolic induction and restriction functors, and studying the corresponding monads acting on categories of cuspidal objects. It is hoped that the same techniques will be fruitful in understanding similar questions in the group, elliptic, mirabolic, quantum, and modular settings.

Keywords

D-modules Geometric representation theory Springer theory 

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Notes

Acknowledgements

Much of this paper was written while I was a postdoc at MSRI, and I would like to thank them for their hospitality. This work (which arose from my PhD thesis) has benefited greatly from numerous conversations with various people over the last few years. The following is a brief and incomplete list of such individuals whom I would especially like to thank, with apologies to those who are omitted. P. Achar, G. Bellamy, D. Ben-Zvi, D. Fratila, D. Gaitsgory, D. Jordan, D. Juteau, P. Li, C. Mautner, D. Nadler, L. Rider, T. Schedler. I am particularly grateful to an anonymous referee for their comments and suggestions on an earlier draft.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasAustinUSA
  2. 2.School of MathematicsUniversity of EdinburghEdinburghUK

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