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Generalized Springer theory for \(D\)-modules on a reductive Lie algebra. (English) Zbl 1422.20015

Sel. Math., New Ser. 24, No. 5, 4223-4277 (2018); correction ibid. 27, No. 4, Paper No. 62, 10 p. (2021).
Summary: 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.

MSC:

20G05 Representation theory for linear algebraic groups
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
17B45 Lie algebras of linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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