Représentation de Weil et $\beta $-extensions

Blondel, Corinne

Annales de l'Institut Fourier, Tome 62 (2012), p. 1319-1366 / Harvested from Numdam

Résumé

Nous étudions les $\beta $-extensions dans un groupe classique $p$-adique et obtenons une relation entre certaines $\beta $-extensions à l’aide d’une représentation de Weil. Nous en donnons une application à l’étude des points de réductibilité de certaines induites paraboliques.

Abstract

We study $\beta $-extensions in a $p$-adic classical group and we produce a relation between some $\beta $-extensions by means of a Weil representation. We apply this to the study of reducibility points of some parabolically induced representations.

Détail

Publié le : 2012-01-01

MR 3025745 | Zbl 1263.22010

DOI : https://doi.org/10.5802/aif.2724
Classification:  22E50
Mots clés: Corps local non archimédien, groupe classique, représentation de Weil, beta-extension, type semi-simple, caractère semi-simple, paire couvrante, algèbre de Hecke, points de réductibilité.

Bibliographie

[1] Blasco, L.; Blondel, C. Algèbres de Hecke et séries principales généralisées de $Sp_4(F)$, Proc. London Math. Soc., Tome 85(3) (2002), pp. 659-685 | Article | MR 1936816 | Zbl 1017.22010

[2] Blondel, C. Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N), Ann. scient. Ec. Norm. Sup., Tome 37 (2004), pp. 533-558 | Numdam | MR 2097892 | Zbl 1063.22016

[3] Blondel, C. Covers and propagation in symplectic groups, Univ. Aarhus (Functional analysis IX, Various Publ. Ser. (Aarhus)) Tome 48 (2007), pp. 16-31 | MR 2349437 | Zbl 1139.22013

[4] Blondel, C.; Stevens, S. Genericity of supercuspidal representations of $p$-adic $\text{Sp}4$, Compositio Math., Tome 145(1) (2009), pp. 213-246 | Article | MR 2480501 | Zbl 1217.22013

[5] Bushnell, C.J.; Kutzko, P.C. The admissible dual of $\text{GL}(N)$ via compact open subgroups, Annals of Mathematics Studies 129, Princeton (1993) | MR 1204652 | Zbl 0787.22016

[6] Bushnell, C.J.; Kutzko, P. Smooth representations of reductive $p$-adic groups : structure theory via types, Proc. London Math. Soc., Tome 77 (1998), pp. 582-634 | Article | MR 1643417 | Zbl 0911.22014

[7] Bushnell, C.J.; Kutzko, P. Semisimple types in $\text{GL}_n$, Compositio Math., Tome 119 (1999), pp. 53-97 | Article | MR 1711578 | Zbl 0933.22027

[8] Gan, W.T.; Takeda, S. The Local Langlands Conjecture for $Sp(4)$, Int. Math. Res. Not., Tome 2010 (2010), pp. 2987-3038 | MR 2673717 | Zbl 1239.11061

[9] Gérardin, P. Weil representations associated to finite fields, J. of Algebra, Tome 46 (1977), pp. 54-101 | Article | MR 460477 | Zbl 0359.20008

[10] Goldberg, D.; Kutzko, P.; Stevens, S. Covers for self-dual supercuspidal representations of the Siegel Levi subgroup of classical p-adic groups, Int. Math. Res. Not., Tome 2007 (2007) | MR 2376206 | Zbl 1133.22010

[11] Howlett, R.; Lehrer, G. Induced cuspidal representations and generalised Hecke rings, Invent. Math., Tome 58 (1980), pp. 37-64 | Article | MR 570873 | Zbl 0435.20023

[12] Jantzen, C. Discrete series for $p$-adic $\text{SO}(2n) $ and restrictions of representations of $ {\mathrm{O}}(2n)$ (to appear in Canadian Journal of Mathematics (2011)) | Zbl 1219.22016

[13] Kutzko, P.; Morris, L. Level zero Hecke algebras and parabolic induction : the Siegel case for split classical groups, Int. Math. Res. Not., Tome 2006 (2006) | MR 2276353 | Zbl 1115.22013

[14] Mœglin, C. Normalisation des opérateurs d’entrelacement et réductibilité des induites de cuspidales ; le cas des groupes classiques p-adiques, Ann. of Math., Tome 151 (2000), pp. 817-847 | Article | MR 1765711 | Zbl 0956.22012

[15] Morris, L. Tamely ramified intertwining algebras, Ann. of Math., Tome 114 (1993), pp. 1-54 | MR 1235019 | Zbl 0854.22022

[16] Neuhauser, M. An explicit construction of the metaplectic representation over a finite field, J. of Lie Theory, Tome 12 (2002), pp. 15-30 | MR 1885034 | Zbl 1026.22018

[17] Shahidi, F. A proof of Langlands conjecture on Plancherel measure ; complementary series for p-adic groups, Ann. of Math., Tome 132 (1990), pp. 273-330 | Article | MR 1070599 | Zbl 0780.22005

[18] Silberger, A. Special representations of reductive p-adic groups are not integrable, Ann. of Math., Tome 111 (1980), pp. 571-587 | Article | MR 577138 | Zbl 0437.22015

[19] Stevens, S. Double coset decomposition and intertwining, manuscripta math., Tome 106 (2001), pp. 349-364 | Article | MR 1869226 | Zbl 0988.22008

[20] Stevens, S. Semisimple characters for $p$-adic classical groups, Duke Math. J., Tome 127(1) (2005), pp. 123-173 | Article | MR 2126498 | Zbl 1063.22018

[21] Stevens, S. The supercuspidal representations of $p$-adic classical groups, Invent. Math., Tome 172 (2008), pp. 289-352 | Article | MR 2390287 | Zbl 1140.22016

[22] Szechtman, F. Weil representations of the symplectic group, J. of Algebra, Tome 208 (1998), pp. 662-686 | Article | MR 1655472 | Zbl 0932.20047

[23] Zhang, Y. Discrete series of classical groups, Canad. J. Math., Tome 52(5) (2000), pp. 1101-1120 | Article | MR 1782340 | Zbl 0961.22013