Higher ramification and the local Langlands correspondence

Colin J. Bushnell; Guy Henniart

doi:https://doi.org/10.4007/annals.2017.185.3.5

Abstract

Let $F$ be a non-Archimedean locally compact field. We show that the local Langlands correspondence over $F$ 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 ${\rm GL}_n(F)$ 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 $\Psi_\varTheta$ depending on the endo-class $\varTheta$ of $\theta_\pi$. We give a method for calculating $\Psi_\varTheta$ directly from $\varTheta$. Consequently, the ramification-theoretic structure of $\sigma$ can be predicted from the simple character $\theta_\pi$ alone.

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@ARTICLE{2, mrkey = {1423022}, issn = {0073-8301}, author = {Bushnell, Colin J. and Henniart, Guy}, mrclass = {11S37 (22E50)}, url = {http://www.numdam.org/item?id=PMIHES_1996__83__105_0}, journal = {Inst. Hautes Études Sci. Publ. Math.}, zblnumber = {0878.11042}, volume = {83}, mrnumber = {1423022}, fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, mrreviewer = {A. Deitmar}, title = {Local tame lifting for {${\rm GL}(N)$}. {I}. {S}imple characters}, year = {1996}, pages = {105--233}, doi = {10.1515/9781400882496}, }
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@BOOK{3, mrkey = {1685898}, issn = {0303-1179}, author = {Bushnell, Colin J. and Henniart, Guy}, mrclass = {11S37 (11F70 22E50)}, series = {Astérisque}, zblnumber = {0920.11079}, volume = {254}, mrnumber = {1685898}, fjournal = {Astérisque}, mrreviewer = {Marko TadiÄ�}, title = {Local Tame Lifting for {${\rm GL}(n)$}. {II}. {W}ildly Ramified Supercuspidals}, year = {1999}, pages = {vi+105}, }
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Higher ramification and the local Langlands correspondence

Abstract

Authors