Local parameters of supercuspidal representations.

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, GenestierLafforgue and Fargues-Scholze have attached a semisimple parameter Lss to each irreducible representation π. Our first result shows that the Genestier-Lafforgue parameter of a tempered π can be uniquely refined to a tempered L-parameter L(π), thus giving the unique local Langlands correspondence which is compatible with the GenestierLafforgue construction. Our second result establishes ramification properties of Lss (π) for unramified G and supercuspidal π constructed by induction from an open compact (modulo center) subgroup. If Lss is pure in an appropriate sense, we show that Lss (π) is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show Lss 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 P1 and a simple application of Deligne’s Weil II. [ABSTRACT FROM AUTHOR]