Comptes Rendus
Generalized Verma modules, b-functions of semi-invariants and duality for twisted 𝒟-modules on generalized flag manifolds
Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 111-116.

Let G be a connected semisimple algebraic group over , 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.

Soient G un groupe algébrique sur 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.

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DOI: 10.1016/S1631-073X(02)02449-4
Corrado Marastoni 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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Corrado Marastoni. Generalized Verma modules, b-functions of semi-invariants and duality for twisted $ \mathcal{D}$-modules on generalized flag manifolds. Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 111-116. doi : 10.1016/S1631-073X(02)02449-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02449-4/

[1] A. D'Agnolo; P. Schapira Leray's quantization of projective duality, Duke Math. J., Volume 84 (1996) no. 2, pp. 453-496

[2] A. Gyoja Highest weight modules and b-functions of semi-invariants, Publ. Res. Inst. Math. Sci., Volume 30 (1994) no. 3, pp. 353-400

[3] M. Kashiwara b-functions and holonomic systems, Invent. Math., Volume 38 (1976/77), pp. 33-53

[4] M. Kashiwara The Riemann–Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 2, pp. 319-365

[5] M. Kashiwara Representation theory and 𝒟-modules on flag varieties, Astérisque, Volume 173–174 (1989), pp. 55-109

[6] C. Marastoni Grassmann duality for 𝒟-modules, Ann. Sci. École Norm. Sup., Volume 31 (1998), pp. 459-491

[7] C. Sabbah Proximité évanescente II, Compositio Math., Volume 64 (1987), pp. 213-241

[8] M. Sato; M. Kashiwara; T. Kimura; T. Oshima Microlocal analysis of prehomogeneous vector spaces, Invent. Math., Volume 62 (1980), pp. 117-179

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