Local Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$-factors
Local Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$-factors
复制标题
$mathrm{GL}_{n}$ 以及外部和对称平方 $varepsilon$ 因子的局部 Langlands 对应关系
DOI：
10.1215/00127094-2017-0001
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发表时间：
2014
影响因子：
2.5
通讯作者：
Tung
中科院分区：
文献类型：
--
作者：
J. Cogdell;F. Shahidi;Tung
设$F$是$p$-adic域，即，一个有限的扩展$\mathbb Q_p$对于某个素数$p$。局部朗兰兹对应附加到$W '_F$的每个连续$n$-维$\Phi$-半单表示$\rho$、$\bar F/F$的Weil-Deligne群、$GL_n（F）$的不可约容许表示$\pi（\rho）$，使得除其他外，对的局部$L$-和$\varepp $-因子被保持。这种对应关系应该是健壮的，并在算术和分析方面保留各种并行操作，例如取外正方形或对称正方形。在本文中，我们证明了这是局部算术和解析对称平方和外平方$\vareps $-因子的情况，即$\vareps（s，\Lambda ^2\rho，\psi）=\vareps（s，\pi（\rho），\Lambda ^2，\psi）$和$\vareps（s，Sym^2\rho，\psi）=\vareps（s，\pi（\rho），Sym^2，\psi）$。$L$-函数的一致性也遵循我们的方法，但这是Henniart已经知道的。证明是一个强大的变形参数，结合局部/全局技术，这减少了问题的稳定性分析$\gamma$-因子$\gamma（s，\pi，\Lambda ^2，\psi）$下高度分歧扭曲时，$\pi$是超尖点。这最后一步是通过有关的$\gamma$-因子的梅林变换的部分贝塞尔函数附加到表示，然后分析的渐近性的部分贝塞尔函数，启发部分由理论的沙利卡芽贝塞尔积分。每个不可约的容许表示$\pi$的稳定性然后从相应的算术$\gamma$-因子作为推论。
Let $F$ be a $p$--adic field, i.e., a finite extension of $\mathbb Q_p$ for some prime $p$. The local Langlands correspondence attaches to each continuous $n$--dimensional $\Phi$-semisimple representation $\rho$ of $W'_F$, the Weil--Deligne group for $\bar F/F$, an irreducible admissible representation $\pi(\rho)$ of $GL_n(F)$ 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 paper, 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 $\varepsilon(s,Sym^2\rho,\psi)=\varepsilon(s,\pi(\rho),Sym^2,\psi)$. 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.