Intégrales orbitales sur \(GL(N)\) et corps locaux proches. (Orbital integrals on \(GL(N)\) and close local fields.). (French) Zbl 0853.22012

Summary: Let \(F\) be a local non-archimedean field, \(N\) an integer \(\geq 2\), \(\underline{G}=GL(N)\), \(n\) a positive integer and \({\mathcal H}(\underline{G}(F),K_{F}^{n})\) the Hecke algebra of \(\underline{G}(F)\) with respect to the congruence subgroup modulo \({\mathcal P}_{F}^{ n}\) of \(\underline{G}({\mathcal O}_{F})\). We prove an explicit formula for the elliptic orbital integrals of functions in \({\mathcal H}(\underline{G}(F),K_{F}^{n})\). Thanks to this formula, for \(\gamma \in \underline{G}(F)\) semi-simple regular, we produce an integer \(r = r(\gamma,n)\geq n\) such that for any local non-archimedean field \(F'\) \(r\)-close to \(F\) (i.e. such that there exists an isomorphism of rings \({\mathcal O}_{F}/{\mathcal P}_{F}^{ r}\simeq {\mathcal O}_{F'}/ {\mathcal P}_{F'}^{ r}\)), there exists \(\gamma' \in \underline{G}(F')\) semi-simple regular such that the orbital integrals at \(\gamma\) of all functions in \({\mathcal H}(\underline{G}(F),K_{F}^{n})\) match, via a given isomorphism of algebras \({\mathcal H}(\underline{G}(F),K_{F}^{n})\simeq {\mathcal H}(\underline{G}(F'),K_{F'}^{n})\), those of functions in \({\mathcal H}(\underline{G}(F'),K_{F'}^{n})\) at \(\gamma'\).


22E50 Representations of Lie and linear algebraic groups over local fields
Full Text: DOI Numdam EuDML


[1] [B] , Éléments de mathématique : Algèbre VIII, Hermann, Paris, 1958. · Zbl 0102.27203
[2] [BK] , , The admissible dual of GL(N) via compact open subgroups, Ann. of Math. Studies, 129, Princeton Univ. Press, Princeton, N.J., 1993. · Zbl 0787.22016
[3] [Bo] , Linear algebraic groups (second enlarged edition), Graduate Texts in Math., 126, Springer, New York, 1991. · Zbl 0726.20030
[4] [C] , Orbital integrals on p-adic groups : a proof of the Howe conjecture, Ann. of Math., 129 (1989), 237-251. · Zbl 0675.22007
[5] [D] , Les corps locaux de caractéristique p, limites de corps locaux de caractéristique zéro, in Représentations des groupes réductifs sur un corps local, Hermann, Coll. Travaux en cours, Paris (1984), 119-157. · Zbl 0578.12014
[6] [HC1] , Harmonic analysis on reductive p-adic groups, Lectures Notes in Math., 162, Springer-Verlag, Berlin-Heidelberg-New York, 1970. · Zbl 0202.41101
[7] [HC2] , A submersion principle and its applications in Papers dedicated to the memory of V.K. Patodi, Indian Academy of Sciences, Bangalore, and the Tata Institute of Fundamental Research, Bombay (1980), 95-102 (Collected papers IV, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1984, 439-446).
[8] [HH] , , Automorphic induction for GL(n) (over local non-archimedean fields), prépublication U. de Paris-Sud, à paraître dans Duke Math. J. (mais sans l’Appendix 1). · Zbl 0849.11092
[9] [Ho] , Harish-Chandra homomorphism for p-adic groups, Regional Conferences Series in Math., 59 (1985), Amer. Math. Soc., Providence, R.I. · Zbl 0593.22014
[10] [K] , Representations of groups over close local fields, J. Analyse Math., 47 (1986), 175-179. · Zbl 0634.22010
[11] [La] , Cohomology with compact supports of Drinfeld modular varieties, Cambridge Univ. Press, 1991.
[12] [Le] , Thèse, Univ. de Paris-Sud, 8 février 1994.
[13] [S] , Introduction to the arithmetic theory of automorphic functions, Princeton U. Press, Princeton, 1971. · Zbl 0221.10029
[14] [V] , Caractérisation des intégrales orbitales sur un groupe réductif p-adique, J. Fac. Sci. U. of Tokyo, 28 (3) Sec. 14 (1982), 945-961. · Zbl 0499.22011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.