Biquantization techniques for computing characters of differential operators on Lie groups

Panagiotis Batakidis

doi:10.1016/j.crma.2010.11.012

Lie Algebras/Functional Analysis

Biquantization techniques for computing characters of differential operators on Lie groups
[Techniques de bi-quantification pour le calcul des caractères des opérateurs différentiels sur les groupes de Lie]

Présenté par : Michel Duflo

Panagiotis Batakidis ¹

¹ Department of Mathematics, Aristotle University of Thessaloniki, Konstantinou Karamanli, Panepistimioupoli, 54124 Thessaloniki, Greece

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 247-250.

Résumé
Abstract

Nous comparons le caractère de lʼalgèbre $(U (g) / U {(g) h_{λ})}^{h}$ , tel quʼutilisé par Fujiwara et Corwin–Greenleaf, avec le caractère produit par les techniques de bi-quantification appliquées au cas des algèbres de Lie par Cattaneo–Torossian. Nous démontrons que ces deux caractères coïncident, à une algèbre (de spécialisation) plus petite près. Nous discutons également un exemple bien connu et nous obtenons des informations supplémentaires quant à la question de savoir si la symétrisation est un isomorphisme dʼalgèbres.

We compare the character of the algebra $(U (g) / U {(g) h_{λ})}^{h}$ , as used by Fujiwara and Corwin and Greenleaf, with the character produced from biquantization techniques applied in the Lie case by Cattaneo and Torossian. We prove that up to a smaller (specialization) algebra, these two characters are the same. An old example is also treated and it is proved that we now get more information about the question of when the symmetrization is an isomorphism of algebras.

Reçu le : 2010-09-23
Accepté le : 2010-11-10
Publié le : 2010-12-30

DOI : 10.1016/j.crma.2010.11.012

Affiliations des auteurs :

Panagiotis Batakidis ¹

¹ Department of Mathematics, Aristotle University of Thessaloniki, Konstantinou Karamanli, Panepistimioupoli, 54124 Thessaloniki, Greece

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     author = {Panagiotis Batakidis},
     title = {Biquantization techniques for computing characters of differential operators on {Lie} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {247--250},
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     doi = {10.1016/j.crma.2010.11.012},
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Panagiotis Batakidis. Biquantization techniques for computing characters of differential operators on Lie groups. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 247-250. doi : 10.1016/j.crma.2010.11.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.012/

Bibliographie
Cité par

[1] P. Batakidis, Déformation par quantification et théorie de Lie, Thèse de Doctorat, Univ. Paris VII, 2009.

[2] P. Batakidis Deformation quantization and invariant differential operators, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 3–4, pp. 143-148

[3] Y. Benoist Analyse harmonique sur les espaces symètriques nilpotents, J. Funct. Anal., Volume 59 (1984), pp. 211-254

[4] A.S. Cattaneo; G. Felder Relative formality theorem and quantization of coisotropic submanifolds, Adv. Math., Volume 208 (2007) no. 2, pp. 521-548

[5] A.S. Cattaneo; B. Keller; Ch. Torossian; A. Bruguières Déformation, quantization, théorie de Lie, Collection Panoramas et Synthèse, vol. 20, SMF, 2005

[6] A.S. Cattaneo; Ch. Torossian Quantification pour les paires symmétriques et diagrammes de Kontsevich, Ann. Sci. Ecole Norm. Sup. (5) (2008), pp. 787-852

[7] L.J. Corwin; F.P. Greenleaf Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity, Comm. Pure Appl. Math., Volume 45 (1992) no. 6, pp. 681-748

[8] H. Fujiwara Analyse harmonique pour certaines représentations induites dʼun groupe de Lie nilpotent, J. Math. Soc. Japan, Volume 50 (1998) no. 3, pp. 753-766

[9] H. Fujiwara; G. Lion; B. Magneron; S. Mehdi A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces, Math. Ann., Volume 327 (2003) no. 3, pp. 513-544

[10] M. Kontsevich Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216

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