Regular trace formula and base change for GL(n)
Flicker, Yuval Z.
Annales de l'Institut Fourier, Tome 40 (1990), p. 1-30 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

La formule de trace “régulière” pour une fonction avec une composante locale qui est bi-invariante par un sous-groupe d’Iwahori et suffisamment régulière par rapport aux autres composantes, est développée dans le contexte d’un groupe réductif. On l’utilise pour donner une démonstration élémentaire de la théorie des changements des bases pour les représentations automorphes cuspidales de GL(n) qui ont une composante supercuspidale. On donne aussi une preuve locale du transfert des intégrales orbitales de fonctions locales générales, qui est au contraire obtenu comme corollaire de la correspondance locale.

The “regular”trace formula, for a test function with a local component which is Iwahori-biinvariant and sufficiently regular with respect to the other components, is developed in the context of a reductive group. It is used to give a simple proof of the theory of base-change for cuspidal automorphic representations of GL(n) which have a supercuspidal component. A purely local proof is given to transfer orbital integrals of sufficiently many spherical functions, by relating them to regular Iwahori functions. Transfer of orbital integrals of smooth functions is not used in the proof. Instead it is obtained as a corollary to the local lifting.

@article{AIF_1990__40_1_1_0,
     author = {Flicker, Yuval Z.},
     title = {Regular trace formula and base change for $GL(n)$},
     journal = {Annales de l'Institut Fourier},
     volume = {40},
     year = {1990},
     pages = {1-30},
     doi = {10.5802/aif.1201},
     mrnumber = {91d:11139},
     zbl = {0691.10018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1990__40_1_1_0}
}

Flicker, Yuval Z. Regular trace formula and base change for $GL(n)$. Annales de l'Institut Fourier, Tome 40 (1990) pp. 1-30. doi : 10.5802/aif.1201. http://gdmltest.u-ga.fr/item/AIF_1990__40_1_1_0/

[AC] J. Arthur, L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Annales of Math. Study, 120 (1989). | Zbl 0682.10022

[BDK] J. Bernstein, P. Deligne, D. Kazhdan, Trace Paley-Wiener theorem, J. Analyse Math., 47 (1986), 180-192. | MR 88g:22016 | Zbl 0634.22011

[BDKV] J. Bernstein, P. Deligne, D. Kazhdan, M. F. Vigneras, Représentations des groupes réductifs sur un corps local, Hermann, Paris, 1984. | Zbl 0544.00007

[BZ] J. Bernstein, A. Zelevinsky, Induced representation of reductive p-adic groups I, Ann. Scient. Ecole Norm. Sup., 10 (1977), 441-472. | Numdam | Zbl 0412.22015

[B] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math., 35 (1976), 233-259. | MR 56 #3196 | Zbl 0334.22012

[BJ] A. Borel, H. Jacquet, Automorphic forms and automorphic representations, Proc. Symp. Pure Math., 33 (1979), I, 189-208. | MR 81m:10055 | Zbl 0414.22020

[C] P. Cartier, Representations of p-adic groups: A survey, Proc. Symp. Pure Math., 33 (1979), I 111-155. | MR 81e:22029 | Zbl 0421.22010

[CD] W. Casselman, Characters and Jacquet modules, Math. Ann., 230 (1977), 101-105; see also: P. Deligne, Le support du caractère d'une représentation supercuspidale, CRASP, 283 (1976), 155-157. | MR 58 #11237 | Zbl 0337.22019

[D] V. Drinfeld, Elliptic modules II, Mat. Sbornik, 102 (144) (1977). (2) (= Math. USSR Sbornik, 31 (1977)(2), 159-170). | MR 55 #12644 | Zbl 0386.20022

[F1] Y. Flicker, Rigidity for automorphic forms, J. Analyse Math., 49 (1987), 135-202. | MR 89a:11057 | Zbl 0656.10024

[F2] Y. Flicker, Regular trace formula and base change lifting, Amer. J. Math., 110 (1988), 739-764. | MR 89m:11051 | Zbl 0666.10020

[F3] Y. Flicker, The trace formula and base-change for GL(3), SLN 927 (1982). | MR 84d:10035 | Zbl 0481.10023

[F4] Y. Flicker, Stable base-change for spherical functions, Nagoya Math. J., 106 (1987), 121-142. | MR 89b:11098 | Zbl 0616.22005

[F5] Y. Flicker, On the symmetric-square : Total global comparison; preprint, Harvard, 1986.

[F6] Y. Flicker, Base change trace identity for U(3), J. Analyse Math., 52 (1989), 39-52. | MR 90m:11080 | Zbl 0684.10027

[FK] Y. Flicker and D. Kazhdan, Metaplectic correspondence, Publ. Math. IHES, 64 (1987), 53-110. | Numdam | MR 88d:11049 | Zbl 0616.10024

[FK1] Y. Flicker and D. Kazhdan, A simple trace formula, J. Analyse Math., 50 (1988), 189-200. | MR 90e:11078 | Zbl 0666.10018

[FK2] Y. Flicker and D. Kazhdan, Geometric Ramanujan conjecture and Drinfeld's reciprocity law, in Number Theory, Trace Formulas and Discrete Groups, Selberg Symposium, Oslo, June (1987), Academic Press (1988), 201-218. | MR 90g:11063 | Zbl 0674.10025

[JS] H. Jacquet, J. Shalika, On Euler products and the classification of automorphic forms II, Amer. J. Math., 103 (1981), 777-815. | MR 82m:10050b | Zbl 0491.10020

[K] D. Kazhdan, Cuspidal geometry of p-adic groups, J. Analyse Math., 47 (1986), 1-36. | MR 88g:22017 | Zbl 0634.22009

[KL] D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math., 87 (1987), 153-215. | MR 88d:11121 | Zbl 0613.22004

[Ko] R. Kottwitz, Base change for unit elements of Hecke algebras, Compos. Math., 60 (1986), 237-250. | Numdam

[L] R. Langlands, Base change for GL(2), Annals of Math. Study, 96 (1980). | MR 82a:10032 | Zbl 0444.22007

[R] J. Rogawski, Trace-Paley-Wiener theorem in the twisted case, Trans. AMS, 309 (1988), 215-229. | MR 89k:22035 | Zbl 0663.22011

[S] J. P. Serre, Corps locaux, Hermann, Paris, 1968.

[Z] A. Zelevinski, Induced representations of reductive p-adic groups II. On irreducible representations of GL(n), Ann. Scient. Ecole Norm. Sup., 13 (1980), 165-210. | Numdam | MR 83g:22012 | Zbl 0441.22014