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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.

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p > 0 . Using Poincaré series, we globalize supercuspidal representations of H_F in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands–Shahidi \gamma -factors and the local Langlands correspondence for classical groups.

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.