Hamiltonian reduction and nearby cycles for mirabolic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">D</mml:mi></mml:math>-modules

Bellamy, Gwyn; Ginzburg, Victor

doi:10.1016/j.aim.2014.10.002

Cited by 12 publications

(22 citation statements)

References 36 publications

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“…The group GL n acts on X by the adjoint action on the first factor, and the standard representation on the second factor. We consider the category of GL n -equivariant D-modules on X; more generally, we can consider the category of c-monodromic D-modules for any c P C. There is a well known relationship between such D-modules and modules for the spherical subalgebra of the rational Cherednik algebra (with parameter c); these ideas have been the subject of considerable interest, for notably in the work of Ginzburg with Bellamy, Etingof, Finkelberg, and Gan [BG15,EG02,FG10,GG06]. Our results in this paper suggest a new approach to this topic via parabolic induction and restriction functors.…”

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confidence: 99%

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

Gunningham

2018

Sel. Math. New Ser.

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Given a reductive group G, we give a description of the abelian category of Gequivariant D-modules on g " LiepGq, 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 pL, Eq, consisting of a Levi subgroup L, and a cuspidal local system E on a nilpotent L-orbit. Each block is equivalent to the category of D-modules on the center zplq of l which are equivariant for the action of the relative Weyl group N G pLq{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.Main results. In his seminal paper [Lus84], Lusztig proved the Generalized Springer Correspondence, which gives a description of the category of G-equivariant perverse sheaves on the nilpotent cone N G Ď g " LiepGq, for a reductive group G:The sum is indexed by cuspidal data: pairs pL, Eq of a Levi subgroup L of G and simple cuspidal local system on a nilpotent orbit for L, up to simultaneous conjugacy. For each such Levi L, W G,L " N G pLq{L denotes the corresponding relative Weyl group.The main result of this paper is that Lusztig's result extends to a description of the abelian category Mpgq G of all G-equivariant D-modules on g:Theorem A. There is an equivalence of abelian categories:where the sum is indexed by cuspidal data pL, Eq.Here zplq denotes the center of the Lie algebra l of a Levi subgroup L which carries an action of the finite group W G,L , 1 and Mpzplqq WG,L denotes the category of W G,L -equivariant D-modules on zplq, or equivalently, modules for the semidirect product D zplq¸WG,L . If we restrict to the subcategory of modules with support on the nilpotent cone (which can be identified with the category of equivariant 1 In fact, in the cases when L carries a cuspidal local system, W G,L is a Coxeter group and zplq its reflection representation.

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confidence: 99%

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

Gunningham

2018

Sel. Math. New Ser.

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“…Objects of QCoh(D X , G) are known as mirabolic D-modules. A certain localization of this category is closely related to representations of the rational Cherednik algebra and the geometry of the Hilbert scheme of points in the plane (see, e.g., [11], [8], [2]). • Given a quiver Q = (Q 1 , Q 0 ) and fixing a dimension vector α ∈ N Q0 gives rise to another example of a vector space…”

Section: Motivation: Quantization Of Cotangent Stacksmentioning

confidence: 99%

Projective Generation for Equivariant -Modules

Bellamy

Gunningham

Raskin

2021

Transformation Groups

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We investigate compact projective generators in the category of equivariant "Image missing"-modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X, there is a natural countable set of compact projective generators indexed by finite dimensional representations of G. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest.

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“…Similarly, we say that M is -monodromic if , where is the differential of the action on M . These notions can be defined sheaf theoretically; see [2, Definition 2.3.2]. Note that monodromic -modules are automatically weakly equivariant.…”

Section: Hamiltonian Reduction and A Mirabolic Generalizationmentioning

confidence: 99%

“…As shown in [2], the case where c is not generic is much more interesting. (In particular, there the category of mirabolic sheaves need not be semisimple, and we expect not to be semisimple when the category is not.)…”

Section: Hamiltonian Reduction and A Mirabolic Generalizationmentioning

confidence: 99%

Filtrations on Springer fiber cohomology and Kostka polynomials

Bellamy

Schedler

2017

Lett Math Phys

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We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

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Hamiltonian reduction and nearby cycles for mirabolic D-modules

Cited by 12 publications

References 36 publications

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

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

Projective Generation for Equivariant -Modules

Filtrations on Springer fiber cohomology and Kostka polynomials

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