Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T23:49:43.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable base change for spherical functions

Published online by Cambridge University Press:  22 January 2016

Yuval Z. Flicker
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let E/F be an unramified cyclic extension of local non-archimedean fields, G a connected reductive group over F, K(F) (resp. K(E)) a hyper-special maximal compact subgroup of G(F) (resp. G(E)), and H(F) (resp. H(E)) the Hecke convolution algebra of compactly-supported complex-valued K(F) (resp. G(E))-biinvariant functions on G(F) (resp. G(E)). Then the theory of the Satake transform defines (see § 2) a natural homomorphism H(E) → H(F), θ→f. There is a norm map N from the set of stable twisted conjugacy classes in G(E) to the set of stable conjugacy classes in G(F); it is an injection (see [Ko]). Let Ω‱(x, f) denote the stable orbital integral of f in H(F) at the class x, and Ω‱(y, θ) the stable twisted orbital integral of θ in H(E) at the class y.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 106 , June 1987 , pp. 121 - 142
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[A] Arthur, J., On a family of distributions obtained from Eisenstein series II: explicit formulas, Amer. J. Math., 104 (6) (1982), 1289–1336.Google Scholar
[AC] Arthur, J., Clozel, L., Base change for GL(n), manuscript (1987).Google Scholar
[BZ] Bernstein, I., Zelevinskii, A., Representations of the group GL(n,F) where F is a non-archimedean local field, Uspekhi Mat. Nauk, 31 (3) (1976), 5–70.Google Scholar
[BZ′] Bernstein, I., Induced representations of reductive p-adic groups I, Ann. Sci Ecole Norm Sup., 10 (1977), 441–472.Google Scholar
[B] Borel, A., Automorphic L-functions, Proc. Sympos. Pure Math., 33 (1979), II 27–63.Google Scholar
[B′] Borel, A., Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math., 35 (1976), 233–259.CrossRefGoogle Scholar
[BJ] Borel, A., Jacquet, H., Automorphic forms and automorphic representations, Proc. Sympos. Pure Math., 33 (1979), I 189–202.Google Scholar
[Ca] Cartier, P., Representations of p-adic groups: A survey, Proc. Sympos. Pure Math., 33 (1979), I 111–155.CrossRefGoogle Scholar
[C] Casselman, W., Characters and Jacquet modules, Math. Ann., 230 (1977), 101–105; see also: Deligne, P., Le support du caractère d’une représentation super-cuspidal, CRASP, 283 (1976), 155–157.Google Scholar
[D] van Dijk, G., Computations of certain induced characters of p-adic groups, Math. Ann., 199 (1972), 229–240.Google Scholar
[U(2)] Flicker, Y., Stable and labile base change for 17(2), Duke Math. J., 49 (1982), 691–729.Google Scholar
[GL(3)] Flicker, Y., The trace formula and base change for GL(3), Springer Lecture Notes, 927 (1982).Google Scholar
[GL(n)] Flicker, Y., On twisted lifting, Trans. Amer. Math. Soc, 290 (1985), 161–178.Google Scholar
[UP] Flicker, Y., Unitary quasi-lifting; preparations, Contemp. Math., 53 (1986), 119–139.Google Scholar
[UA] Flicker, Y., Unitary quasi-lifting: applications, Trans. Amer. Math. Soc, 294 (1986), 553–565.Google Scholar
[Rig] Flicker, Y., Rigidity for automorphic forms: I. Harmonic analysis; II. Comparison; III. Representations of simple algebras; IV. Automorphic forms on compact unitary groups; J. d’Analyse Math., 48 (1987).Google Scholar
[Sym] Flicker, Y., On the Symmetric Square; I. Definitions and lemmas; II. Applications of a trace formula; III. Orbital integrals, Math. Ann., (1987); IV. Twisted trace formula; V. Unstable local transfer (with D. Kazhdan); VI. Total global comparison; preprints, Harvard 1986.Google Scholar
[RTF] Flicker, Y., Regular trace formula and base-change lifting, Amer. J. Math., (1988).Google Scholar
[FK] Flicker, Y. and Kazhdan, D., Metaplectic correspondence, Publ. Math. IHES, 64 (1987), 53–110.Google Scholar
[STF] Flicker, Y. and Kazhdan, D., A simple trace formula, J. d’Analyse Math., 48 (1987).Google Scholar
[G] Greenberg, M., Schemata over local rings II, Ann. of Math., 78 (1963), 256–266.CrossRefGoogle Scholar
[K] Kazhdan, D., Cuspidal geometry on p-adic groups, J. d’Analyse Math., 47 (1987), 1–36.Google Scholar
[K′] Kazhdan, D., On lifting, in Springer Lecture Notes, 1041 (1984), 209–249.Google Scholar
[Ko′] Kottwitz, R., Rational conjugacy classes in reductive groups, Duke Math. J., 49 (1982), 785–806.Google Scholar
[Ko′] Kottwitz, R., Base change for unit elements of Hecke algebras, Compositio, Math., 60 (1986), 237–250.Google Scholar
[R] Rao, R., Orbital integrals in reductive groups, Ann. of Math., 96 (1972), 505–510.CrossRefGoogle Scholar
[Se] Serre, J. P., Corps locaux, Hermann, Paris.Google Scholar
[Ta] Tate, J., The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J., 27 (1966), 709–719.CrossRefGoogle Scholar
[Ti] Tits, J., Reductive groups over local fields, Proc. Sympos. Pure Math., 33 (1979), I, 29–69.Google Scholar