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We establish the existence and uniqueness of twisted exterior and symmetric square γ-factors in positive characteristic by studying the Siegel Levi case of generalized spinor groups. The corresponding theory in characteristic zero is due to Shahidi. In addition, in characteristic p we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic p, we obtain a proof of the stability property of γ-factors under twists by highly ramified characters. Next we use the results on the compatibility of the Langlands-Shahidi local coefficients with the Deligne-Kazhdan theory over close local fields to show that the twisted symmetric and exterior square γ-factors, L-functions and ε-factors are preserved over close local fields. Furthermore, we obtain a formula for Plancherel measures in terms of local factors and we also show that they also preserved over close local fields.

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Abstract. 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. Contrary to the local proof given by Shahidi, our proof uses a refined version of the local-global principle for GLn in positive characteristic, which has independent interest. The comparison of γ-factors is made via a uniqueness result for Rankin-Selberg γ-factors over a non-Archimedean local field of positive characteristic.