On the generalized Ramanujan and Arthur conjectures over function fields

Journal article

Let G be a simple group over a global function field K, and let π 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 πu is tempered and πv is unramified and generic. We prove that πw is tempered at all unramified places Kw 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 πv is in the set corresponding to the nilpotent class N, and if πu satisfies an analogous hypothesis, then πw 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 π 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 π. 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.

Authors: Ciubotaru, D; Harris, M

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: Princeton University
• Journal: Annals of Mathematics
• Acceptance date: 2025-10-09
• EISSN: 1939-8980
• ISSN: 0003-486X
• Language: English