## Duke Mathematical Journal

### Local Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$-factors

#### Abstract

Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb{Q}_{p}$ for some prime $p$. The local Langlands correspondence (LLC) attaches to each continuous $n$-dimensional $\Phi$-semisimple representation $\rho$ of $W'_{F}$, the Weil–Deligne group for $\overline{F}/F$, an irreducible admissible representation $\pi(\rho)$ of $\mathrm{GL}_{n}(F)$ such that, among other things, the local $L$- and $\varepsilon$-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 $\varepsilon$-factors, that is, that $\varepsilon(s,\Lambda^{2}\rho,\psi)=\varepsilon(s,\pi(\rho),\Lambda^{2},\psi)$ and $\varepsilon(s,\operatorname{Sym}^{2}\rho,\psi)=\varepsilon(s,\pi(\rho),\operatorname{Sym}^{2},\psi)$. 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 $\gamma$-factor $\gamma(s,\pi,\Lambda^{2},\psi)$ under highly ramified twists when $\pi$ is supercuspidal. This last step is achieved by relating the $\gamma$-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 $\pi$ then follows from those of the corresponding arithmetic $\gamma$-factors as a corollary.

#### Article information

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

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

https://projecteuclid.org/euclid.dmj/1493863448

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

Mathematical Reviews number (MathSciNet)
MR3694565

Zentralblatt MATH identifier
06775427

#### 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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