Changements de base explicites des représentations supercuspidales de $U(1,1)\left(F_0\right)$

Blasco, Laure¹ 

¹ Université Paris-Sud Département de Mathématiques U.M.R. 8628 du C.N.R.S. Bâtiment 425 91405 Orsay cedex (France)

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 905-938.

Résumé (VO)

Soit $F_0$ un corps local non archimédien de caractéristique nulle et de caractéristique résiduelle impaire. On décrit explicitement les changements de base des représentations supercuspidales de $U(1,1)\left(F_0\right)$. C’est une étape vers la description du changement de base des paquets endoscopiques supercuspidaux de $U(2,1)\left(F_0\right)$.

Abstract

Let $F_0$ be a nonarchimedean local field of characterisitic 0 and odd residual characteristic. We describe explicitly the two base change lifts of supercuspidal representations of $U(1,1)\left(F_0\right)$. This represents a step towards the goal of describing base change of endoscopic supercuspidal $L$-packets of $U(2,1)\left(F_0\right)$.

Détail

DOI : 10.5802/aif.2542

Classification : 22E50, 11F70
Mots-clés : corps local, changement de base, groupe unitaire, représentations supercuspidales, $L$-paquets
Keywords: Local field, base change, unitary group, supercuspidal representations, $L$-packets

Bibliographie

[1] Adler, J.; Lansky, J. Depth-zero base change for unramified $U(2,1)$, J. Number Theory, Volume 114 (2005), pp. 324-360 | DOI | Zbl | MR

[2] Adler, J.; Lansky, J. Depth-zero base change for ramified $U(2,1)$, 2008 (arXiv :0807.1528v1)

[3] Blasco, L. Description du dual admissible de $U(2,1)\left(F\right)$ par la théorie des types de C. Bushnell et P. Kutzko, Manuscripta Math., Volume 107 (2002), pp. 151-186 | DOI | Zbl | MR

[4] Blasco, L. Types, paquets et changement de base : l’exemple de $U(2,1)\left(F_0\right)$, I, Canadian J. Math., Volume 60 (2008), pp. 790-821 | DOI | Zbl | MR

[5] Bushnell, C.; Henniart, G. Local tame lifting for GL$\left(N\right)$ I : simple characters, Publ. Math. IHES, Volume 83 (1996), pp. 105-233 | Zbl | MR | mathdoc-id

[6] Bushnell, C.; Henniart, G. Local tame lifting for GL$\left(n\right)$ II : wildly ramified supercuspidals, Astérisque, Soc. Math. France, Volume 254 (1999), pp. vi+105 | Zbl | MR | mathdoc-id

[7] Bushnell, C.; Henniart, G. The local Langlands conjecture for $\mathrm{GL}\left(2\right)$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Springer-Verlag, Berlin, 2006 | Zbl | MR

[8] Ennola, V. On the characters of the finite unitary groups, Ann. Acad. Scien. Fenn., Volume 323 (1963), pp. 1-35 | Zbl | MR

[9] Flicker, Y. Stable and labile base change for $U\left(2\right)$, Duke Math. J., Volume 49 (1982), pp. 691-729 | DOI | Zbl | MR

[10] Flicker, Y. On distinguished representations, J. Reine Angew. Math., Volume 418 (1991), pp. 134-172 | Zbl | MR

[11] Fröhlich, A. Principal orders and embedding of local fields in algebras, Proc. London Math. Soc., Volume 54 (1987), pp. 247-266 | DOI | Zbl | MR

[12] Hakim, J.; Murnaghan, F. Two types of distinguished supercuspidal representations, Int. Math. Res. Not., Volume 35 (2002), pp. 1857-1889 | DOI | Zbl | MR

[13] Jacquet, H.; Langlands, R. Automorphic forms on $\mathrm{GL}\left(2\right)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970 | Zbl | MR

[14] Kottwitz, R. Rational conjugacy classes on reductive groups, Duke Math. J., Volume 49 (1982), pp. 785-806 | DOI | Zbl | MR

[15] Kutzko, P. C.; Sally, P. J. Jr. All supercuspidal representations of $S{L}_{\ell }$ over a $p$-adic field are induced, Representation theory of reductive groups (Park City, Utah, 1982) (Progr. Math.), Volume 40, Birkhäuser, Boston, MA, 1983, pp. 185-196 | Zbl

[16] Labesse, J.-P.; Langlands, R. $L$-indistinguishability for $SL\left(2\right)$, Canadian J. Math., Volume 31 (1979), pp. 726-785 | DOI | Zbl | MR

[17] Morris, L. Tamely ramified supercuspidal representations, Ann. scient. Éc. Norm. Sup., Volume 29 (1996), pp. 639-667 | Zbl | MR | mathdoc-id

[18] Rogawski, J. Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, 123, Princeton University Press, Princeton, NJ, 1990 | Zbl | MR

[19] Springer, T. A. Characters of special groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) (Lecture Notes in Mathematics, Vol. 131), Springer, Berlin, 1970, pp. 121-166 | Zbl | MR

[20] Stevens, S. Double coset decompositions and intertwining, Manuscripta Math., Volume 106 (2001), pp. 349-364 | DOI | Zbl | MR