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

Part of: Discontinuous groups and automorphic forms Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  20 December 2012

Eyal Kaplan
Eyal Kaplan*
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel (email: [email protected])
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Abstract

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.

Keywords

Rankin–Selberg integralsgcd$\uppercase {l}$-functions

MSC classification

Secondary: 11S40: Zeta functions and $L$-functions 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 4 , April 2013 , pp. 587 - 636
Copyright
Copyright © 2012 The Author(s)

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