  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 $\pi_w$ is tempered at all unramified
places $K_w$ at which $G$ is unramified quasi-split.
  More generally, the set of unitary spherical representations is partitioned
according to nilpotent conjugacy classes in the Lie algebra of $G$. We show
that if $\pi_v$ is in the set corresponding to the nilpotent class $N$, and if
$\pi_u$ satisfies an analogous hypothesis, then $\pi_w$ belongs to the same
class $N$, where $w$ is as above. These results are consistent with conjectures
of Shahidi and Arthur.
  The proofs use 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 unitary spherical representations, due to Barbasch and the
first-named author, the theory of weights excludes almost all 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.

Number Theory;Representation Theory