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On the gcd of local Rankin–Selberg integrals for even orthogonal groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 4 / April 2013
- Published online by Cambridge University Press:
- 20 December 2012, pp. 587-636
- Print publication:
- April 2013
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- Article
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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.
