On categories of equivariant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">D</mml:mi></mml:math>-modules

Lörincz, András; Walther, Uli

doi:10.1016/j.aim.2019.04.051

Cited by 15 publications

(18 citation statements)

References 57 publications

(124 reference statements)

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“…Remark 3. The abelian category QCoh(D X , G) and the induced modules P X (V ) are also extensively studied in a paper by Lőrincz and Walther [17]. The questions studied in that paper are of a somewhat different nature to those in the present paper.…”

Section: The Equivariant Derived Categorymentioning

confidence: 83%

“…-is non-zero; -is indecomposable; -is holonomic, or has a holonomic direct summand or submodule. Some results in this direction have been obtained by Lőrincz and Walther [17]. For example, it is shown in Theorem 3.22 of loc.…”

Section: Further Questionsmentioning

confidence: 88%

“…• In the case where G = T acts linearly on a vector space X, localizations of the category QCoh(D X , G) are related to quantizations of hypertoric varieties (see, e.g., [6]). • Many more interesting examples can be found in [17].…”

Section: Motivation: Quantization Of Cotangent Stacksmentioning

confidence: 99%

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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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Section: The Equivariant Derived Categorymentioning

confidence: 83%

Section: Further Questionsmentioning

confidence: 88%

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Projective Generation for Equivariant -Modules

Bellamy

Gunningham

Raskin

2021

Transformation Groups

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“…In this paper we complete the analogous study for the rest of the representations within the series (1.1), emphasizing the uniformity of the methods and results. Further, this completes a necessary step toward the classification of such objects on irreducible representations of reductive groups with finitely many orbits that has been initiated through several articles [21,23,24,25,26,31,32,33,36], see also [14] and references therein for their Bernstein-Sato polynomials.…”

Section: Introductionmentioning

confidence: 88%

Local cohomology on a subexceptional series of representations

Lörincz¹,

Weyman²

2023

Annales De l'Institut Fourier

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We consider a series of four subexceptional representations coming from the third line of the Freudenthal-Tits magic square; using Bourbaki notation, these are representations (G , X) corresponding to (C 3 , ω 3 ), (A 5 , ω 3 ), (D 6 , ω 5 ), and (E 7 , ω 6 ). In each case X has five G = G × C-orbits, displaying some uniform behavior, e.g. their dimensions or defining ideals. In this paper, we determine some further invariants and analyze their uniformity within the series. We describe the category of G-equivariant coherent D X -modules as the category of representations of a quiver. We construct explicitly the simple equivariant D-modules and describe their G-structures. We determine the D-module structure of local cohomology modules supported in orbit closures, and calculate intersection cohomology groups and Lyubeznik numbers. While our results for (A 5 , ω 3 ), (D 6 , ω 5 ), (E 7 , ω 6 ) are still completely uniform, the case (C 3 , ω 3 ) displays a surprisingly different behavior, for which we give two explanations: the middle orbit is not simply-connected, and its closure is not Gorenstein.Résumé. -Nous considérons une série de quatre représentations sous-exceptionnelles venant de la troisième ligne du carré magique de Freudenthal-Tits: (G , X) = (C 3 , ω 3 ), (A 5 , ω 3 ), (D 6 , ω 5 ), ou (E 7 , ω 6 ), en utilisant la notation de Bourbaki. Dans chaque cas, X a cinq G = G × C-orbites, qui se révèlent avoir un comportement uniforme, avec par exemple leurs dimensions ou leurs idéaux définissants. Dans cet article, nous obtenons plus d'invariants et nous étudions leur uniformité dans cette série des représentations. Nous décrivons la catégorie des D X -modules cohérentes G-équivariantes, et décrivons leurs G-structures. Nous déterminons, pour les modules de cohomologie locale avec support dans des clôtures d'orbites, leur structure comme D-module, et calculons des groupes de cohomologie d'intersection et des nombres de Lyubeznik. Alors que nos résultats pour (A 5 , ω 3 ), (D 6 , ω 5 ), (E 7 , ω 6 ) sont uniformes, le cas (C 3 , ω 3 ) fait apparaître un comportement différent et unique, pour lequel nous donnons deux explications : l'orbite moyenne n'est pas simplement connexe, et sa clôture n'est pas Gorenstein.

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“…Since all orbit stabilizers are connected, [HTT08, Theorem 11.6.1] shows that each orbit can only support one equivariant local system, the constant one. See [LW18] for more details on equivariant D-modules.…”

Section: A Recursionmentioning

confidence: 99%

Weight filtrations on GKZ-systems

Reichelt¹,

Walther²

2018

Preprint

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If β ∈ C d is integral but not a strongly resonant parameter for the homogeneous matrix A ∈ Z d×n with ZA = Z d , then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here in the normal case the corresponding weight filtration by computing the intersection complexes with respective multiplicities on the associated graded parts. We do this by computing the weight filtration of a Gauss-Manin system with respect to a locally closed embedding of a torus inside an affine space. Our results show that these data, which we express in terms of intersection cohomology groups on induced toric varieties, are purely combinatorial, and not arithmetic, in the sense that they only depend on the polytopal structure of the cone of A but not on the semigroup itself. As a corollary we get a purely combinatorial formula for the length of the underlying (regular) holonomic D-module, irrespective of homogeneity. In dimension up to three, and for simplicial semigroups, we give explicit generators of the weight filtration.

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On categories of equivariant D-modules

Cited by 15 publications

References 57 publications

Projective Generation for Equivariant -Modules

Projective Generation for Equivariant -Modules

Local cohomology on a subexceptional series of representations

Weight filtrations on GKZ-systems

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