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CALCULS DE FACTEURS EPSILON DE PAIRES POUR GLn SUR UN CORPS LOCAL, I

Published online by Cambridge University Press:  01 September 1999

COLIN J. BUSHNELL
Affiliation:
Department of Mathematics, King's College, Strand, London WC2R 2LS
GUY HENNIART
Affiliation:
URA 752 du CNRS, Université de Paris-Sud, 91405 Orsay cedex, France
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Abstract

Soient F un corps commutatif localement compact non archimédien et ψ un caractère additif non trivial de F. Soient σ une représentation du groupe de Weil–Deligne de F, et ψσˇ sa contragrédiente. Nous calculons le facteur ε(σ[otimes ]σˇ, ψ, ½). De manière analogue, nous calculons le facteur ε(π×πˇ, ψ, ½) pour toute représentation admissible irréductible π de GLn(F). En conséquence, si F est de caractéristique nulle et si σ et π se correspondent par la correspondance de Langlands construite par M. Harris, ou celle construite par les auteurs, alors les facteurs ε(σ[otimes ]σˇ, ψ, s) et ε(π×πˇ, ψ, s) sont égaux pour tout nombre complexe s.

Let F be a non-Archimedean local field and ψa non-trivial additive character of F. Let σ be a representation of the Weil–Deligne group of F and σˇ its contragredient representation. We compute ε(σ[otimes ]σˇ, ψ, ½). Analogously, we compute ε(π×πˇ, ψ, ½) for all irreducible admissible representations π of GLn(F). Consequently, if F has characteristic zero, and σ, π correspond via the Langlands correspondence established by M. Harris or the correspondence constructed by the authors, then we have ε(σ[otimes ]σˇ, ψ, s) = ε(π×πˇ, ψ, s) for all s∈[Copf ].

Type
NOTES AND PAPERS
Copyright
© The London Mathematical Society 1999

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