Abstract
Let $F$ be a non-Archimedean locally compact field. We show that the local Langlands correspondence over $\mathbb{F}$ has a property generalizing the higher ramification theorem of local class field theory. If $\pi$ is an irreducible cuspidal representation of a general linear group $\mathrm{GL}_n(F)$ and $\sigma$ the corresponding irreducible representation of the Weil group $\mathcal{W}_F$ of $F$, the restriction of $\sigma$ to a ramification subgroup of $\mathcal{W}_F$ is determined by a truncation of the simple character $\theta_\pi$ contained in $\pi$, and conversely. Numerical aspects of the relation are governed by an Herbrand-like function $\Psi_\Theta$ depending on the endo-class $\Theta$ of $\theta_\pi$. We give a method for calculating $\Psi_\Theta$ directly from $\Theta$. Consequently, the ramification-theoretic structure of $\sigma$ can be predicted from the simple character $\theta_\pi$ alone.
Citation
Colin Bushnell. Guy Henniart. "Higher ramification and the local Langlands correspondence." Ann. of Math. (2) 185 (3) 919 - 955, May 2017. https://doi.org/10.4007/annals.2017.185.3.5
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