A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

Maxim Gurevich

doi:10.5802/crmath.384

Théorie des représentations

A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

Maxim Gurevich ¹

¹ Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 21-30.

Résumé
Abstract

Dans le cas des représentations Iwahori-sphériques de groupes généraux linéaires de corps non archimédien, Chan-Savin ont récemment obtenu une expression du foncteur de Whittaker comme restriction d’un module d’algèbre d’Iwahori–Hecke finie à une composante isotypique. Nous généralisons cette méthode pour décrire les foncteurs de Whittaker dégénérés principaux. Parallèlement, nous interprétons la formule de Murnaghan pour le caractère de Harish-Chandra–Howe comme une expansion du groupe de Grothendieck de ce même module.

En comparant les deux approches selon le prisme des algèbres PSH de Zelevinsky, nous obtenons une matrice de transition unitriangulaire explicite entre les coefficients d’expansion du caractère et les dimensions de Whittaker dégénérées principales.

For Iwahori-spherical representations of non-Archimedean general linear groups, Chan–Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori–Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan’s formula for the Harish-Chandra–Howe character as a Grothendieck group expansion of the same module.

Comparing the two approaches through the lens of Zelevinsky’s PSH-algebras, we obtain an explicit unitriangular transition matrix between coefficients of the character expansion and the principal degenerate Whittaker dimensions.

Reçu le : 2022-05-09
Révisé le : 2022-05-24
Accepté le : 2022-05-25
Publié le : 2023-01-12

DOI : 10.5802/crmath.384

Affiliations des auteurs :

Maxim Gurevich ¹

¹ Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A triangular system for local character expansions of {Iwahori-spherical} representations of general linear groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {21--30},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.384},
     language = {en},
}

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Maxim Gurevich. A triangular system for local character expansions of Iwahori-spherical representations of general linear groups. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 21-30. doi : 10.5802/crmath.384. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.384/

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