Local parameters of supercuspidal representations

For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(\pi)$ 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 $\CL(\pi)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(\pi)$ for unramfied $G$ and supercuspidal $\pi$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(\pi)$ is pure in an appropriate sense, we show that $\CL^{ss}(\pi)$ is ramified (unless $G$ is 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 Poincar\'e series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II.