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On the gcd of local Rankin–Selberg integrals for even orthogonal groups

Published online by Cambridge University Press:  20 December 2012

Eyal Kaplan*
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel (email: kaplaney@post.tau.ac.il)
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Abstract

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We study the Rankin–Selberg integral for a pair of representations of ${\rm SO}_{2l}\times {\rm GL}_{n}$, where ${\rm SO}_{2l}$ is defined over a local non-Archimedean field and is either split or quasi-split. The integrals span a fractional ideal, and its unique generator, which contains any pole which appears in the integrals, is called the greatest common divisor (gcd) of the integrals. We describe the properties of the gcd and establish upper and lower bounds for the poles. In the tempered case we can relate it to the $L$-function of the representations defined by Shahidi. Results of this work may lead to a gcd definition for the $L$-function.

Type
Research Article
Copyright
Copyright © 2012 The Author(s)

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