Local parameters of supercuspidal representations

Let $G$ be a connected reductive group over the non-archime\-dean local field $F$ and let $\pi$ be a supercuspidal representation of $G(F)$. The local Langlands conjecture posits that to such a $\pi$ can be attached a parameter $L(\pi)$, which is an equivalence class of homomorphisms from the Weil-Deligne group with values in the Langlands $L$-group ${}^LG$ over an appropriate algebraically closed field $C$ of characteristic $0$. When $F$ is of positive characteristic $p$ then Genestier and Lafforgue have defined a parameter, $L^{ss}(\pi)$, which is a homomorphism $W_F \ra {}^LG(C)$ that is {\it semisimple} in the sense that, if the image of $L^{ss}(\pi)$, intersected with the Langlands dual group $\hat{G}(C)$, is contained in a parabolic subgroup $P \subset \hat{G}(C)$, then it is contained in a Levi subgroup of $P$. If the Frobenius eigenvalues of $L^{ss}(\pi)$ are pure in an appropriate sense, then the local Langlands conjecture asserts that the image of $L^{ss}(\pi)$ is in fact {\it irreducible} -- its image is contained in no proper parabolic $P$. In particular, unless $G = GL(1)$, $L^{ss}(\pi)$ is ramified: it is non-trivial on the inertia subgroup $I_F \subset W_F$. In this paper we prove, at least when $G$ is split and semisimple, that this is the case provided $\pi$ can be obtained as the induction of a representation of a compact open subgroup $U \subset G(F)$, and provided the constant field of $F$ is of order greater than $3$. Conjecturally every $\pi$ is compactly induced in this sense, and the property was recently proved by Fintzen to be true as long as $p$ does not divide the order of the Weyl group of $G$. The proof is an adaptation of the globalization method of \cite{GLo} when the base curve is $\PP^1$, and a simple application of Deligne's Weil II.