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

Gurevich, Maxim

Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 21-30

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.

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.

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

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

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

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

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

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

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

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

[9] Gourevitch, Dmitry; Sahi, Siddhartha 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] Howe, Roger The Fourier transform and germs of characters (case of ${\mathrm{Gl}}_n$ over a $p$-adic field), Math. Ann., Volume 208 (1974), pp. 305-322 | MR | Zbl | DOI

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

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

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

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

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

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

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

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

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

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

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