  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.
