Uniqueness of Rankin-Selberg products

In the present paper, we show the equality of the $γ$-factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands-Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local-global principle for ${\rm GL}_n$ which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of $γ$-factors is made via a uniqueness result for Rankin-Selberg $γ$-factors.

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