Generalized Verma modules, <i>b</i>-functions of semi-invariants and duality for twisted $ \mathcal{D}$-modules on generalized flag manifolds

Marastoni, Corrado

doi:10.1016/S1631-073X(02)02449-4

Generalized Verma modules, b-functions of semi-invariants and duality for twisted

𝒟

-modules on generalized flag manifolds
[Modules de Verma généralisés, b-fonctions des quasi-invariants et dualité pour les

𝒟

-modules tordus sur les variétés de drapeaux généralisées]

Marastoni, Corrado ^1,²

¹ Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni, 7, 35131 Padova, Italy
² RIMS, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

Comptes Rendus. Mathématique, Tome 335 (2002) no. 2, pp. 111-116.

Résumé
Abstract

Soient G un groupe algébrique sur $ℂ$ connexe semi-simple, P un sous-groupe parabolique, $𝔤$ et $𝔭$ leurs algèbres de Lie. On démontre une version microlocale des conjectures de Gyoja [2] sur une relation entre l'irreductibilité des modules de Verma généralisés sur $𝔤$ induits de $𝔭$ et les zéros des b-fonctions des quasi-invariants sur G par rapport à P. Notre méthode utilise une dualité pour les $𝒟$ -modules tordus sur les variétés de drapeaux généralisées.

Let G be a connected semisimple algebraic group over $ℂ$ , P a parabolic subgroup, $𝔤$ and $𝔭$ their Lie algebras. We prove a microlocal version of Gyoja's conjectures [2] about a relation between the irreducibility of generalized Verma modules on $𝔤$ induced from $𝔭$ and the zeroes of b-functions of P-semi-invariants on G. Our method uses a duality for twisted $𝒟$ -modules on generalized flag manifolds.

Reçu le : 2002-04-26
Accepté le : 2002-05-21
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02449-4

Affiliations des auteurs :

Marastoni, Corrado ^{1, 2}

¹ Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni, 7, 35131 Padova, Italy
² RIMS, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

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     title = {Generalized {Verma} modules, \protect\emph{b}-functions of semi-invariants and duality for twisted $ \mathcal{D}$-modules on generalized flag manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {111--116},
     publisher = {Elsevier},
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     number = {2},
     year = {2002},
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     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02449-4/}
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Marastoni, Corrado. Generalized Verma modules, b-functions of semi-invariants and duality for twisted $ \mathcal{D}$-modules on generalized flag manifolds. Comptes Rendus. Mathématique, Tome 335 (2002) no. 2, pp. 111-116. doi : 10.1016/S1631-073X(02)02449-4. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02449-4/

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