Skip to main content Accessibility help
×
Home

On a lifting problem of L-packets

  • Bin Xu (a1)

Abstract

Let $G\subseteq \widetilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of those of $\widetilde{G}$ . Motivated by this, we hope to construct the L-packets of $\widetilde{G}$ from those of $G$ . The primary example in our mind is when $G=\text{Sp}(2n)$ , whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and $\widetilde{G}=\text{GSp}(2n)$ . As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\widetilde{G}$ from those of  $G$ .

Copyright

References

Hide All
[AP06] Adler, J. D. and Prasad, D., On certain multiplicity one theorems , Israel J. Math. 153 (2006), 221245.
[Art13] Arthur, J., The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013).
[Bor79] Borel, A., Automorphic L-functions , in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, OR, 1977), Part 2, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 2761.
[Bou87] Bouaziz, A., Sur les caractères des groupes de Lie réductifs non connexes , J. Funct. Anal. 70 (1987), 179; MR 870753 (89c:22020).
[Clo87] Clozel, L., Characters of non-connected, reductive p-adic groups , Can. J. Math. 39 (1987), 149167.
[Del76] Deligne, P., Les constantes locales de l’équation fonctionnelle de la fonction L d’Artin d’une représentation orthogonale , Invent. Math. 35 (1976), 299316.
[GGP12] Gan, W. T., Gross, B. H. and Prasad, D., Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups , Astérisque 346 (2012), 1109; Sur les conjectures de Gross et Prasad. I.
[GK82] Gelbart, S. S. and Knapp, A. W., L-indistinguishability and R groups for the special linear group , Adv. Math. 43 (1982), 101121.
[Har63] Harish-Chandra, Invariant eigendistributions on semisimple Lie groups , Bull. Amer. Math. Soc. 69 (1963), 117123.
[Har75] Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term , J. Funct. Anal. 19 (1975), 104204.
[Har99] Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, University Lecture Series, vol. 16(American Mathematical Society, Providence, RI, 1999), preface and notes by Stephen DeBacker and Paul J. Sally, Jr.
[HT01] Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton, NJ, 2001), with an appendix by Vladimir G. Berkovich.
[Hen00] Henniart, G., Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique , Invent. Math. 139 (2000), 439455.
[HS12] Hiraga, K. and Saito, H., On L-packets for inner forms of SL n , Mem. Amer. Math. Soc. 215 (2012), vi+97.
[JL14] Jantzen, C. and Liu, B., The generic dual of p-adic split SO 2n and local Langlands parameters , Israel J. Math. 204 (2014), 199260.
[JS03] Jiang, D. and Soudry, D., The local converse theorem for SO (2n + 1) and applications , Ann. of Math. (2) 157 (2003), 743806.
[JS04] Jiang, D. and Soudry, D., Generic representations and local Langlands reciprocity law for p-adic SO(2n + 1) , in Contributions to automorphic forms, geometry, and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 457519.
[Kos78] Kostant, B., On Whittaker vectors and representation theory , Invent. Math. 48 (1978), 101184.
[Kot82] Kottwitz, R. E., Rational conjugacy classes in reductive groups , Duke Math. J. 49 (1982), 785806.
[Kot84] Kottwitz, R. E., Stable trace formula: cuspidal tempered terms , Duke Math. J. 51 (1984), 611650.
[KS99] Kottwitz, R. E. and Shelstad, D., Foundations of twisted endoscopy , Astérisque 55 (1999), vi+190.
[KS12] Kottwitz, R. E. and Shelstad, D., On splitting invariants and sign conventions in endoscopic transfer, Preprint (2012), arXiv:1201.5658.
[Lab85] Labesse, J.-P., Cohomologie, L-groupes et fonctorialité , Compos. Math. 55 (1985), 163184.
[LL79] Labesse, J.-P. and Langlands, R. P., L-indistinguishability for SL(2) , Canad. J. Math. 31 (1979), 726785.
[Lan79] Langlands, R. P., Stable conjugacy: definitions and lemmas , Canad. J. Math. 31 (1979), 700725.
[LS87] Langlands, R. P. and Shelstad, D., On the definition of transfer factors , Math. Ann. 278 (1987), 219271.
[Lap04] Lapid, E. M., On the root number of representations of orthogonal type , Compos. Math. 140 (2004), 274286.
[Lem13] Lemaire, B., Caractères tordus des représentations admissibles , Astérisque, to appear, Preprint (2013), arXiv:1007.3576.
[Liu11] Liu, B., Genericity of representations of p-adic Sp2n and local Langlands parameters , Canad. J. Math. 63 (2011), 11071136.
[Mez13] Mezo, P., Character identities in the twisted endoscopy of real reductive groups , Mem. Amer. Math. Soc. 222 (2013), vi+94.
[MS82] Milne, J. S. and Shih, K.-Y., Conjugates of Shimura varieties , in Hodge cycles, motives and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, Berlin–New York, 1982), 280356.
[Mok14] Mok, C. P., Endoscopic classification of representations of quasi-split unitary groups , Mem. Amer. Math. Soc. 235 (2014), vi+248.
[Mor11] Morel, S., Cohomologie d’intersection des variétés modulaires de Siegel, suite , Compos. Math. 147 (2011), 16711740.
[Ngô10] Ngô, B. C., Le lemme fondamental pour les algèbres de Lie , Publ. Math. Inst. Hautes Études Sci. (111) (2010), 1169.
[Sch13] Scholze, P., The local Langlands correspondence for GL n over p-adic fields , Invent. Math. 192 (2013), 663715.
[Sha90] Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups , Ann. of Math. (2) 132 (1990), 273330; MR 1070599 (91m:11095).
[She12] Shelstad, D., On geometric transfer in real twisted endoscopy , Ann. of Math. (2) 176 (2012), 19191985.
[Tat79] Tate, J., Number theoretic background , in Automorphic forms, representations and L-functions (Proc. Symp. Pure Math., Oregon State University, Corvallis, OR, 1977), Part 2, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 326.
[Wal95] Waldspurger, J.-L., Une formule des traces locale pour les algèbres de Lie p-adiques , J. reine angew. Math. 465 (1995), 4199.
[Wal97] Waldspurger, J.-L., Le lemme fondamental implique le transfert , Compos. Math. 105 (1997), 153236.
[Wal06] Waldspurger, J.-L., Endoscopie et changement de caractéristique , J. Inst. Math. Jussieu 5 (2006), 423525.
[Wal08] Waldspurger, J.-L., L’endoscopie tordue n’est pas si tordue , Mem. Amer. Math. Soc. 194 (2008), x+261.
[Xu15] Xu, B., L-packets of quasisplit $\text{GSp}(2n)$ and $\text{GO}(2n)$ , Preprint (2015), arXiv:1503.04897.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

On a lifting problem of L-packets

  • Bin Xu (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed