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.
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Maxim Gurevich 1

@article{CRMATH_2023__361_G1_21_0, author = {Maxim Gurevich}, 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}, }
TY - JOUR AU - Maxim Gurevich TI - A triangular system for local character expansions of Iwahori-spherical representations of general linear groups JO - Comptes Rendus. Mathématique PY - 2023 SP - 21 EP - 30 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.384 LA - en ID - CRMATH_2023__361_G1_21_0 ER -
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/
[1] 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] Local character expansions, Ann. Sci. Éc. Norm. Supér., Volume 30 (1997) no. 5, pp. 553-567 | DOI | MR | Zbl
[3] Representations of reductive
[4] Smooth representations of reductive
[5] Iwahori component of the Gelfand-Graev representation, Math. Z., Volume 288 (2018) no. 1-2, pp. 125-133 | DOI | MR | Zbl
[6] The Wavefront Sets of Iwahori-Spherical Representations of Reductive
[7] Representations of finite groups of Lie type, London Mathematical Society Student Texts, 95, Cambridge University Press, 2020 (second edition of [111884].) | DOI | Zbl
[8] Generalized and degenerate Whittaker models, Compos. Math., Volume 153 (2017) no. 2, pp. 223-256 | DOI | MR | Zbl
[9] 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] Admissible invariant distributions on reductive
[11] The Fourier transform and germs of characters (case of
[12] A note on degenerate Whittaker models for general linear groups, J. Number Theory, Volume 209 (2020), pp. 212-224 | DOI | MR | Zbl
[13] Models of representations and Langlands functoriality, Can. J. Math., Volume 72 (2020) no. 3, pp. 676-707 | DOI | MR | Zbl
[14] Sur l’involution de Zelevinski, J. Reine Angew. Math., Volume 372 (1986), pp. 136-177 | MR | Zbl
[15] Modèles de Whittaker dégénérés pour des groupes
[16] Level zero
[17] Unrefined minimal
[18] Local character expansions of admissible representations of
[19] The Weyr characteristic, Am. Math. Mon., Volume 106 (1999) no. 10, pp. 919-929 | DOI | MR | Zbl
[20] Sur les germes de Shalika pour les groupes linéaires, Math. Ann., Volume 284 (1989) no. 2, pp. 199-221 | DOI | Zbl
[21] Induced representations of reductive
[22] Representations of finite classical groups. A Hopf algebra approach, Lecture Notes in Mathematics, 869, Springer, 1981 | DOI | MR | Zbl
- Quasi-admissible, raisable nilpotent orbits, and theta representations, Science China Mathematics (2025) | DOI:10.1007/s11425-024-2359-6
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