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- W4223923892 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." @default.
- W4223923892 created "2022-04-19" @default.
- W4223923892 creator A5051705029 @default.
- W4223923892 creator A5054820772 @default.
- W4223923892 date "2022-04-12" @default.
- W4223923892 modified "2023-09-26" @default.
- W4223923892 title "On the generalized Ramanujan conjecture over function fields" @default.
- W4223923892 doi "https://doi.org/10.48550/arxiv.2204.06053" @default.
- W4223923892 hasPublicationYear "2022" @default.
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