Propagation de paires couvrantes dans les groupes symplectiques

Blondel, Corinne

doi:10.1090/S1088-4165-06-00295-0

Propagation de paires couvrantes dans les groupes symplectiques
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Represent. Theory 10 (2006), 399-434 Request permission

Abstract:

Let $\pi$ be a self-dual supercuspidal representation of $GL(N,F)$ and $\rho$ a supercuspidal representation of $Sp(2k,F)$, with $F$ a local nonarchimedean field of odd residual characteristic. Given a type, indeed a $Sp(2N+2k,F)$-cover, for the inertial class $[GL(N,F) \times Sp(2k,F), \pi \otimes \rho ]_{Sp(2N+2k,F)}$ satisfying suitable hypotheses, we produce a type, indeed a $Sp(2tN+2k,F)$-cover, for the inertial class $[GL(N,F)^{\times t} \times Sp(2k,F), \pi ^{\otimes t } \otimes \rho ]_{Sp(2tN+2k,F)}$, for any positive integer $t$. We describe the corresponding Hecke algebra as a convolution algebra over an affine Weyl group of type $\tilde C_t$ with quadratic relations inherited from the case $t=1$ and the structural data for $\pi$.

References

Laure Blasco and Corinne Blondel, Algèbres de Hecke et séries principales généralisées de $\textrm {Sp}_4(F)$, Proc. London Math. Soc. (3) 85 (2002), no. 3, 659–685 (French). MR 1936816, DOI 10.1112/S0024611502013667
Corinne Blondel, Critère d’injectivité pour l’application de Jacquet, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 11, 1149–1152 (French, with English and French summaries). MR 1490115, DOI 10.1016/S0764-4442(97)83544-6
Corinne Blondel, Quelques propriétés des paires couvrantes, Math. Ann. 331 (2005), no. 2, 243–257 (French). MR 2115455, DOI 10.1007/s00208-004-0579-1
Corinne Blondel, $\textrm {SP}(2N)$-covers for self-contragredient supercuspidal representations of $\textrm {GL}(N)$, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 4, 533–558 (English, with English and French summaries). MR 2097892, DOI 10.1016/j.ansens.2003.10.003
Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
Colin J. Bushnell, Hereditary orders, Gauss sums and supercuspidal representations of $\textrm {GL}_N$, J. Reine Angew. Math. 375/376 (1987), 184–210. MR 882297, DOI 10.1515/crll.1987.375-376.184
Colin J. Bushnell and Philip C. Kutzko, The admissible dual of $\textrm {GL}(N)$ via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993. MR 1204652, DOI 10.1515/9781400882496
Colin J. Bushnell and Philip C. Kutzko, Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. MR 1643417, DOI 10.1112/S0024611598000574
Colin J. Bushnell and Philip C. Kutzko, Types in reductive $p$-adic groups: the Hecke algebra of a cover, Proc. Amer. Math. Soc. 129 (2001), no. 2, 601–607. MR 1712937, DOI 10.1090/S0002-9939-00-05665-3
David Goldberg, Reducibility of induced representations for $\textrm {Sp}(2n)$ and $\textrm {SO}(n)$, Amer. J. Math. 116 (1994), no. 5, 1101–1151. MR 1296726, DOI 10.2307/2374942
Lawrence Morris, Tamely ramified intertwining algebras, Invent. Math. 114 (1993), no. 1, 1–54. MR 1235019, DOI 10.1007/BF01232662
Paul J. Sally Jr. and Marko Tadić, Induced representations and classifications for $\textrm {GSp}(2,F)$ and $\textrm {Sp}(2,F)$, Mém. Soc. Math. France (N.S.) 52 (1993), 75–133 (English, with English and French summaries). MR 1212952
Shaun Stevens, Intertwining and supercuspidal types for $p$-adic classical groups, Proc. London Math. Soc. (3) 83 (2001), no. 1, 120–140. MR 1829562, DOI 10.1112/plms/83.1.120
Marko Tadić, Square integrable representations of classical $p$-adic groups corresponding to segments, Represent. Theory 3 (1999), 58–89. MR 1698200, DOI 10.1090/S1088-4165-99-00071-0
J.-L. Waldspurger, Algèbres de Hecke et induites de représentations cuspidales, pour $\textrm {GL}(N)$, J. Reine Angew. Math. 370 (1986), 127–191 (French). MR 852514, DOI 10.1515/crll.1986.370.127