Duke Mathematical Journal

Local Langlands correspondence for GLn and the exterior and symmetric square ε-factors

J. W. Cogdell, F. Shahidi, and T.-L. Tsai

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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.

Article information

Source
Duke Math. J., Volume 166, Number 11 (2017), 2053-2132.

Dates
Received: 11 May 2015
Revised: 26 September 2016
First available in Project Euclid: 4 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1493863448

Digital Object Identifier
doi:10.1215/00127094-2017-0001

Mathematical Reviews number (MathSciNet)
MR3694565

Zentralblatt MATH identifier
06775427

Subjects
Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 11F80: Galois representations 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]

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

Citation

Cogdell, J. W.; Shahidi, F.; Tsai, T.-L. Local Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$ -factors. Duke Math. J. 166 (2017), no. 11, 2053--2132. doi:10.1215/00127094-2017-0001. https://projecteuclid.org/euclid.dmj/1493863448


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