On the generalized Ramanujan conjecture over function fields

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $\pi_u$ is tempered and $\pi_v$ is unramified and generic. We prove that $\pi$ is tempered at all unramified places $K_w$ at which $G$ is unramified quasi-split. The proof uses the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $\pi$ to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of generic unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes generic complementary series as possible local components of $\pi$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.