Abstract
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 $u$\nonbreakingspace and\nonbreakingspace $v$ (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\nonbreakingspace $K_w$ at which $G$ is unramified quasi-split. \par 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. \par 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.
Full Article