{"title":"Tame multiplicity and conductor for local Galois representations","authors":"C. Bushnell, G. Henniart","doi":"10.2140/tunis.2020.2.337","DOIUrl":null,"url":null,"abstract":"Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $\\sigma$ be an irreducible smooth representation of the absolute Weil group $\\Cal W_F$ of $F$ and $\\sw(\\sigma)$ the Swan exponent of $\\sigma$. Assume $\\sw(\\sigma) \\ge1$. Let $\\Cal I_F$ be the inertia subgroup of $\\Cal W_F$ and $\\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\\varSigma$, of order prime to $p$, so that $\\sigma(\\Cal I_F) = \\sigma(\\Cal P_F)\\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\\sigma$ of any character of $\\varSigma$ is bounded by $\\sw(\\sigma)$.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.337","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2020.2.337","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $\sigma$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(\sigma)$ the Swan exponent of $\sigma$. Assume $\sw(\sigma) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $\sigma(\Cal I_F) = \sigma(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\sigma$ of any character of $\varSigma$ is bounded by $\sw(\sigma)$.