Non-vanishing of geometric Whittaker coefficients for reductive groups

Færgeman, Joakim; Raskin, Sam

doi:10.1090/jams/1051

Non-vanishing of geometric Whittaker coefficients for reductive groups
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by Joakim Færgeman and Sam Raskin;

J. Amer. Math. Soc.

DOI: https://doi.org/10.1090/jams/1051

Published electronically: April 25, 2025

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Abstract:

We prove that cuspidal automorphic $D$-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from $GL_n$ 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 $t$-exact on the category of tempered $D$-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for $GL_n$. We also show that Whittaker coefficient functors are $t$-exact for sheaves with nilpotent singular support. An additional consequence of our results is that the tempered, restricted geometric Langlands conjecture must be $t$-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 $\operatorname {Bun}_G$.

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Non-vanishing of geometric Whittaker coefficients for reductive groups
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Abstract: