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Local Rankin-Selberg convolutions for $\mathrm {GL}_{n}$: Explicit conductor formula


Authors: Colin J. Bushnell, Guy M. Henniart and Philip C. Kutzko
Journal: J. Amer. Math. Soc. 11 (1998), 703-730
MSC (1991): Primary 22E50
DOI: https://doi.org/10.1090/S0894-0347-98-00270-7
MathSciNet review: 1606410
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Abstract: 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.


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Additional Information

Colin J. Bushnell
Affiliation: Department of Mathematics, King’s College, Strand, London WC2R 2LS, United Kingdom
MR Author ID: 43795
Email: bushnell@mth.kcl.ac.uk

Guy M. Henniart
Affiliation: Département de Mathématiques, URA 752 du CNRS, Université de Paris-Sud, 91405 Orsay Cedex, France
MR Author ID: 84385
Email: henniart@dmi.ens.fr or Guy.Henniart@math.u-psud.fr

Philip C. Kutzko
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
MR Author ID: 108580
Email: pkutzko@blue.weeg.uiowa.edu

Keywords: Local field, local constant of pair
Received by editor(s): October 3, 1997
Received by editor(s) in revised form: March 2, 1998
Additional Notes: The first author thanks Université de Paris Sud for hospitality and support for a period during the preparation of this paper.
The research of the second author was partially supported by EU TMR-Network “Arithmetic Geometry and Automorphic Forms” and a joint CNRS, NSF research grant.
The research of the third author was partially supported by NSF grant DMS-9003213 and a joint CNRS, NSF research grant.
Article copyright: © Copyright 1998 American Mathematical Society