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We prove that cuspidal automorphic -modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are -exact on the category of tempered -modules, strengthening a classical result of Gaitsgory (with different hypotheses) for . We also show that Whittaker coefficient functors are -exact for sheaves with nilpotent singular support. An additional consequence of our results is that the tempered, restricted geometric Langlands conjecture must be -exact. We apply our results to show that for suitably irreducible local systems, Whittaker-normalized Hecke eigensheaves are perverse sheaves that are irreducible on each connected component of .
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On the Ramanujan conjecture for automorphic forms over function fields I. Geometry
Let G G be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of G G , subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of Bun G \operatorname {Bun}_G . It is independent of the work of Lafforgue on the global Langlands correspondence.
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- Award ID(s):
- 2001071
- PAR ID:
- 10323301
- Date Published:
- Journal Name:
- Journal of the American Mathematical Society
- Volume:
- 34
- Issue:
- 3
- ISSN:
- 0894-0347
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/jams/968
