Local cohomology on a subexceptional series of representations

Lőrincz, András C.; Weyman, Jerzy

doi:10.5802/aif.3539

Local cohomology on a subexceptional series of representations
[Cohomologie locale sur une série de représentations sous-exceptionnelles]

Lőrincz, András C.¹ ; Weyman, Jerzy ^2,³

¹ David and Judi Proctor Department of Mathematics University of Oklahoma Norman, OK 73019 (USA)
² Department of Mathematics University of Connecticut Storrs, CT 06269 (USA)
³ Instytut Matematyki Uniwersytet Jagielloński Kraków 30-348 (Poland)

Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 747-782.

Résumé
Abstract

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 $G = G^{'} \times ℂ$ -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.

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 $G = G^{'} \times ℂ$ -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.

Reçu le : 2020-10-14
Révisé le : 2021-04-30
Accepté le : 2021-06-23
Publié le : 2023-05-12

DOI : 10.5802/aif.3539

Classification : 14F10, 14B15, 13D45, 13A50, 11S90
Keywords: Local cohomology, equivariant $\mathbb{D}$-modules, prehomogeneous vector spaces.
Mot clés : Cohomologie locale, D-modules équivariants, espaces vectoriels préhomogènes.

Affiliations des auteurs :

Lőrincz, András C. ¹ ; Weyman, Jerzy ^{2, 3}

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Local cohomology on a subexceptional series of representations},
     journal = {Annales de l'Institut Fourier},
     pages = {747--782},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Lőrincz, András C.; Weyman, Jerzy. Local cohomology on a subexceptional series of representations. Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 747-782. doi : 10.5802/aif.3539. https://aif.centre-mersenne.org/articles/10.5802/aif.3539/

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