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/

Bibliographie
Cité par

[1] Dan Barbasch; Allen Moy Whittaker models with an Iwahori fixed vector, Representation theory and analysis on homogeneous spaces(New Brunswick, NJ, 1993) (Simon Gindikin et al., eds.) (Contemporary Mathematics), Volume 177, American Mathematical Society, 1994, pp. 101-105 | DOI | MR | Zbl

[2] Dan Barbasch; Allen Moy Local character expansions, Ann. Sci. Éc. Norm. Supér., Volume 30 (1997) no. 5, pp. 553-567 | DOI | MR | Zbl

[3] Colin J. Bushnell Representations of reductive $p$ -adic groups: localization of Hecke algebras and applications, J. Lond. Math. Soc., Volume 63 (2001) no. 2, pp. 364-386 | DOI | MR | Zbl

[4] Colin J. Bushnell; Philip C. Kutzko Smooth representations of reductive $p$ -adic groups: structure theory via types, Proc. Lond. Math. Soc., Volume 77 (1998) no. 3, pp. 582-634 | DOI | MR | Zbl

[5] Kei Yuen Chan; Gordan Savin Iwahori component of the Gelfand-Graev representation, Math. Z., Volume 288 (2018) no. 1-2, pp. 125-133 | DOI | MR | Zbl

[6] Dan Ciubotaru; Lucas Mason-Brown; Emile Okada The Wavefront Sets of Iwahori-Spherical Representations of Reductive $p$ -adic Groups (2021) (https://arxiv.org/abs/2112.14354)

[7] François Digne; Jean Michel Representations of finite groups of Lie type, London Mathematical Society Student Texts, 95, Cambridge University Press, 2020 (second edition of [111884].) | DOI | Zbl

[8] Raul Gomez; Dmitry Gourevitch; Siddhartha Sahi Generalized and degenerate Whittaker models, Compos. Math., Volume 153 (2017) no. 2, pp. 223-256 | DOI | MR | Zbl

[9] Dmitry Gourevitch; Siddhartha Sahi Generalized and degenerate Whittaker quotients and Fourier coefficients, Representations of reductive groups (Proceedings of Symposia in Pure Mathematics), Volume 101, American Mathematical Society, 2019, pp. 133-154 | DOI | MR | Zbl

[10] Harish-Chandra Admissible invariant distributions on reductive $p$ -adic groups, University Lecture Series, 16, American Mathematical Society, 1999 (with a preface and notes by Stephen DeBacker and Paul J. Sally, Jr.) | Zbl

[11] Roger Howe The Fourier transform and germs of characters (case of ${Gl}_{n}$ over a $p$ -adic field), Math. Ann., Volume 208 (1974), pp. 305-322 | DOI | MR | Zbl

[12] Arnab Mitra A note on degenerate Whittaker models for general linear groups, J. Number Theory, Volume 209 (2020), pp. 212-224 | DOI | MR | Zbl

[13] Arnab Mitra; Eitan Sayag Models of representations and Langlands functoriality, Can. J. Math., Volume 72 (2020) no. 3, pp. 676-707 | DOI | MR | Zbl

[14] Colette Mœglin; Jean-Loup Waldspurger Sur l’involution de Zelevinski, J. Reine Angew. Math., Volume 372 (1986), pp. 136-177 | MR | Zbl

[15] Colette Mœglin; Jean-Loup Waldspurger Modèles de Whittaker dégénérés pour des groupes $p$ -adiques, Math. Z., Volume 196 (1987) no. 3, pp. 427-452 | DOI | Zbl

[16] Lawrence Morris Level zero $G$ -types, Compos. Math., Volume 118 (1999) no. 2, pp. 135-157 | DOI | MR | Zbl

[17] Allen Moy; Gopal Prasad Unrefined minimal $K$ -types for $p$ -adic groups, Invent. Math., Volume 116 (1994) no. 1-3, pp. 393-408 | DOI | MR | Zbl

[18] Fiona Murnaghan Local character expansions of admissible representations of $p$ -adic general linear groups, J. Reine Angew. Math., Volume 554 (2003), pp. 139-155 | DOI | MR | Zbl

[19] Helene Shapiro The Weyr characteristic, Am. Math. Mon., Volume 106 (1999) no. 10, pp. 919-929 | DOI | MR | Zbl

[20] Jean-Loup Waldspurger Sur les germes de Shalika pour les groupes linéaires, Math. Ann., Volume 284 (1989) no. 2, pp. 199-221 | DOI | Zbl

[21] Andrey V. Zelevinsky Induced representations of reductive $p$ -adic groups. II. On irreducible representations of $GL (n)$ , Ann. Sci. Éc. Norm. Supér., Volume 13 (1980) no. 2, pp. 165-210 | DOI | MR | Zbl

[22] Andrey V. Zelevinsky Representations of finite classical groups. A Hopf algebra approach, Lecture Notes in Mathematics, 869, Springer, 1981 | DOI | MR | Zbl

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