Local parameters of supercuspidal representations

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter L^ss(π) 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 Genestier-Lafforgue construction. Our second result establishes ramification properties of L^ss(π) for unramified G and supercuspidal π constructed by induction from an open compact (modulo center) subgroup. If L^ss(π) is pure in an appropriate sense, we show that L^ss(π) is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show L^ss(π) 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.