Local Rankin-Selberg convolutions for 𝐺𝐿_{𝑛}: Explicit conductor formula

Bushnell, Colin ¹   ; Henniart, Guy ²   ; Kutzko, Philip ³

¹ Department of Mathematics, King’s College, Strand, London WC2R 2LS, United Kingdom
² Département de Mathématiques, URA 752 du CNRS, Université de Paris-Sud, 91405 Orsay Cedex, France
³ Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Journal of the American Mathematical Society, Tome 11 (1998) no. 3, pp. 703-730

Résumé

Let $F$ be a non-Archimedean local field and $n_{1}$, $n_{2}$ positive integers. For $i=1,2$, let $G_{i}=\mathrm {GL}_{n_{i}}(F)$ and let $\pi _{i}$ be an irreducible supercuspidal representation of $G_{i}$. Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant $\varepsilon (\pi _{1}\times \pi _{2},s,\psi )$ to the $\pi _{i}$ and an additive character $\psi$ of $F$. This object is of central importance in the study of the local Langlands conjecture. It takes the form \begin{equation*}\varepsilon (\pi _{1}\times \pi _{2},s,\psi ) = q^{-fs}\varepsilon (\pi _{1} \times \pi _{2},0,\psi ), \end{equation*} where $f=f(\pi _{1}\times \pi _{2},\psi )$ is an integer. The irreducible supercuspidal representations of $G=\mathrm {GL}_{n}(F)$ have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of $G$. This paper gives an explicit formula for $f(\pi _{1} \times \pi _{2},\psi )$ in terms of the inducing data for the $\pi _{i}$. It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in $\mathrm {GL}(n)$, developed by Bushnell and Kutzko.

Détail

DOI : 10.1090/S0894-0347-98-00270-7

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