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Suppose $\pi$ and $\rho$ are cuspidal automorphic representations on $GL(n)$ and $GL(m)$ respectively. Then the L-function $L(s,\pi \times \rho)$ has a pole iff and $m=n$ and $\pi$ is isomorphic to the contragradient of $\rho$ by some twist. Does anyone know some reference containing the proof of this fact?

I checked Rankin-Selberg convolution paper by Jacquet-P.S-Shalika. It mentioned this result and said the proof would appear somewhere.

Many thanks.

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I'm also curious, but not so curious right now to pore through the literature. It seems that the J-PS-S paper was followed up by archimidean results by J (Contemp. Math.) and J-S (in the PS 60th bday volume). Perhaps these finish it off? Also, Cogdell's article "Analytic theory of L-functions for GL(n)" (on his webpage) has an exposition of this result. I don't know which of these or other articles is the definitive reference for this important result. – Marty Oct 3 '10 at 18:41
up vote 5 down vote accepted

It seems to be in the Cogdell-PS paper "Remarks on Rankin-Selberg Convolutions" in the Shalika volume, though I haven't read through it myself.

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Since this question just became active again, I thought I'd add this (which surely the OP knew): when $m=n$, this was already in Jacquet-Shalika's "On Euler Products... I" paper, so the only thing needed was the $m < n$ case. – Kimball Jun 25 '15 at 15:59

See also

C. Mœglin and J.-L. Waldspurger, Poles des fonctions L de paires pour GL(N), Appendice, Ann. Sci. École Norm. Sup. (4) 22 (1989), 667–674.

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