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Abstract For a connected reductive groupGover a nonarchimedean local fieldFof positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter$${\mathcal {L}}^{ss}(\pi )$$to each irreducible representation$$\pi $$. Our first result shows that the Genestier-Lafforgue parameter of a tempered$$\pi $$can be uniquely refined to a tempered L-parameter$${\mathcal {L}}(\pi )$$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of$${\mathcal {L}}^{ss}(\pi )$$for unramifiedGand supercuspidal$$\pi $$constructed by induction from an open compact (modulo center) subgroup. If$${\mathcal {L}}^{ss}(\pi )$$is pure in an appropriate sense, we show that$${\mathcal {L}}^{ss}(\pi )$$is ramified (unlessGis a torus). If the inducing subgroup is sufficiently small in a precise sense, we show$$\mathcal {L}^{ss}(\pi )$$is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is$${\mathbb {P}}^1$$and a simple application of Deligne’s Weil II. 

ThetheoryofGaloisrepresentationsattachedtoautomorphicrepresenta- tions of GL(n) is largely based on the study of the cohomology of Shimura varieties of PEL type attached to unitary similitude groups. The need to keep track of the similitude factor complicates notation while making no difference to the final result. It is more natural to work with Shimura varieties attached to the unitary groups themselves, which do not introduce these unnecessary complications; however, these are of abelian type, not of PEL type, and the Galois representations on their cohomology differ slightly from those obtained from the more familiar Shimura varieties. Results on the critical values of the L-functions of these Galois representations have been established by studying the PEL type Shimura varieties. It is not immediately obvious that the automorphic periods for these varieties are the same as for those attached to unitary groups, which appear more naturally in applications of relative trace formulas, such as the refined Gan-Gross-Prasad conjecture (conjecture of Ichino-Ikeda and N. Harris). The present article reconsiders these critical values, using the Shimura varieties attached to unitary groups, and obtains results that can be used more simply in applications.