Waldspurger, J.-L. Sur les germes de Shalika pour les groupes linéaires. (On the Shalika germs for the linear groups). (French) Zbl 0651.22010 Math. Ann. 284, No. 2, 199-221 (1989). Soient F un corps local p-adique, F’ une extension non ramifiée de F de degré f, n’ un entier \(\geq 1\), \(n=n'f\). On considère le groupe \(G=GL(n,F)\) et son sous-groupe \(G'=GL(n',F')\). Soit g’\(\in G'\) un élément elliptique régulier. On montre que si g’ vérifie une certaine condition de régularité, les valeurs au point g’ des germes de Shalika du groupe G sont combinaisons linéaires des valeurs au même point des germes de Shalika du groupe G’. Les coefficients de ces combinaisons linéaires sont liés aux constantes de structure des algèbres de Hall des groupes linéaires sur certains corps finis. On en déduit un calcul par récurrence des valeurs des germes de G sur T, où T est le tore elliptique maximal non ramifié de G. Cela démontre en particulier une formule conjecturée par Rogawski. Reviewer: J.-L.Waldspurger Cited in 2 ReviewsCited in 7 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 11S15 Ramification and extension theory 20G25 Linear algebraic groups over local fields and their integers 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S37 Langlands-Weil conjectures, nonabelian class field theory Keywords:p-adic local field; Shalika germs; maximal elliptic tori PDFBibTeX XMLCite \textit{J. L. Waldspurger}, Math. Ann. 284, No. 2, 199--221 (1989; Zbl 0651.22010) Full Text: DOI EuDML References: [1] Casselman, W.: Introduction to the theory of admissible representations ofp-adic reductive groups. Prépublication [2] Clozel, L.: Orbital integrals onp-adic groups: a proof of the Howe conjecture. Prépublication 1987 [3] Dijk, G. van: Computation of certain induced characters ofp-adic groups. Math. Ann.199, 229-240 (1972) · Zbl 0231.22018 · doi:10.1007/BF01429876 [4] Henniart, G.: On the Langlands conjecture forGL(n): the cyclic case. Ann. Math.123, 145-203 (1986) · Zbl 0588.12010 · doi:10.2307/1971354 [5] Howe, R.: The Fourier transform and germs of characters (case ofGl n over ap-adic field). Math. Ann.208, 305-322 (1974) · Zbl 0273.43011 · doi:10.1007/BF01432155 [6] Kazhdan, D.: Cuspidal geometry ofp-adic groups. J. Anal. Math.47, 1-36 (1986) · Zbl 0634.22009 · doi:10.1007/BF02792530 [7] Kazhdan, D.: On lifting, in Lie group representations. II. (Lecture Notes Mathematics, Vol. 1041) Berlin Heidelberg New York: Springer 1984 · Zbl 0538.20014 [8] Kottwitz, R.: Orbital integrals onGL(3). Am. J. Math.102, 327-384 (1980) · Zbl 0437.22011 · doi:10.2307/2374243 [9] Rodier, F.: Décomposition de la série principale des groupes réductifsp-adiques, in Non commutative harmonic analysis and Lie groups. (Lecture Notes Mathematics, Vol. 880) Berlin Heidelberg New York: Springer 1981 [10] Rogawski, J.: Some remarks on Shalika germs. Contemp. Math.53, 387-391 (1986) · Zbl 0631.22008 [11] Rogawski, J.: An application of the building to orbital integrals. Compos. Math.42, 417-423 (1981) · Zbl 0471.22020 [12] Zelevinsky, A.: Representations of finite classical groups. (Lecture Notes Mathematics, Vol. 869) Berlin Heidelberg New York: Springer 1981 · Zbl 0465.20009 [13] Zelevinsky, A.: Induced representations of reductivep-adic groupsII; on irreducible representations ofGl(n). Ann. Sci. Ec. Norm. Sup.13, 165-210 (1980) · Zbl 0441.22014 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.