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15 August 2017 Local Langlands correspondence for GLn and the exterior and symmetric square ε-factors
J. W. Cogdell, F. Shahidi, T.-L. Tsai
Duke Math. J. 166(11): 2053-2132 (15 August 2017). DOI: 10.1215/00127094-2017-0001

Abstract

Let F be a p-adic field, that is, a finite extension of Qp for some prime p. The local Langlands correspondence (LLC) attaches to each continuous n-dimensional Φ-semisimple representation ρ of W'F, the Weil–Deligne group for F¯/F, an irreducible admissible representation π(ρ) of GLn(F) such that, among other things, the local L- and ε-factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this article, we show that this is the case for the local arithmetic and analytic symmetric square and exterior square ε-factors, that is, that ε(s,Λ2ρ,ψ)=ε(s,π(ρ),Λ2,ψ) and ε(s,Sym2ρ,ψ)=ε(s,π(ρ),Sym2,ψ). The agreement of the L-functions also follows by our methods, but this was already known by Henniart. The proof is a robust deformation argument, combined with local/global techniques, which reduces the problem to the stability of the analytic γ-factor γ(s,π,Λ2,ψ) under highly ramified twists when π is supercuspidal. This last step is achieved by relating the γ-factor to a Mellin transform of a partial Bessel function attached to the representation and then analyzing the asymptotics of the partial Bessel function, inspired in part by the theory of Shalika germs for Bessel integrals. The stability for every irreducible admissible representation π then follows from those of the corresponding arithmetic γ-factors as a corollary.

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J. W. Cogdell. F. Shahidi. T.-L. Tsai. "Local Langlands correspondence for GLn and the exterior and symmetric square ε-factors." Duke Math. J. 166 (11) 2053 - 2132, 15 August 2017. https://doi.org/10.1215/00127094-2017-0001

Information

Received: 11 May 2015; Revised: 26 September 2016; Published: 15 August 2017
First available in Project Euclid: 4 May 2017

zbMATH: 06775427
MathSciNet: MR3694565
Digital Object Identifier: 10.1215/00127094-2017-0001

Subjects:
Primary: 11F66
Secondary: 11F70 , 11F80 , 11S37 , 22E50

Keywords: exterior and symmetric square epsilon factors , local Langlands correspondence , stability of exterior square gamma factors

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 11 • 15 August 2017
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