Représentation de Weil et $\beta$-extensions

Blondel, Corinne

Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1319-1366.

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.

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.

Classification : 22E50
Mot 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.
Keywords: Local non-archimedean field, classical group, Weil representation, beta-extension, semi-simple type, semi-simple character, cover, Hecke algebra, reducibility points.

[1] Blasco, L.; Blondel, C. Algèbres de Hecke et séries principales généralisées de $S{p}_4\left(F\right)$, Proc. London Math. Soc., Volume 85(3) (2002), pp. 659-685 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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., Volume 2007 (2007) | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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