On the generalized Ramanujan conjecture over function fields

TL;DR

This paper proves a generalized Ramanujan conjecture for automorphic representations over function fields, showing that certain local components are tempered based on global conditions, using Galois parametrization and Frobenius weights.

Contribution

It establishes the temperedness of automorphic representations at all unramified places under specific local conditions, extending the Ramanujan conjecture in the function field setting.

Findings

Proves global temperedness from local conditions using Galois parametrization.

Relates Satake parameters to Frobenius weights via Lafforgue's theory.

Excludes complementary series as local components based on weight classification.

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 $u$ 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.