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-normailzed Hecke eigensheaves are perverse
sheaves that are irreducible on each connected component of Bun_G.
Description
Non-vanishing of geometric Whittaker coefficients for reductive groups
%0 Generic
%1 faergeman2022nonvanishing
%A Faergeman, Joakim
%A Raskin, Sam
%D 2022
%K Langlands
%T Non-vanishing of geometric Whittaker coefficients for reductive groups
%U http://arxiv.org/abs/2207.02955
%X 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-normailzed Hecke eigensheaves are perverse
sheaves that are irreducible on each connected component of Bun_G.
@misc{faergeman2022nonvanishing,
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-normailzed Hecke eigensheaves are perverse
sheaves that are irreducible on each connected component of Bun_G.},
added-at = {2022-07-09T08:22:11.000+0200},
author = {Faergeman, Joakim and Raskin, Sam},
biburl = {https://www.bibsonomy.org/bibtex/2a472c93991deae5d681230bb2ebc5da9/dragosf},
description = {Non-vanishing of geometric Whittaker coefficients for reductive groups},
interhash = {6ef55a1387691be3479ab9d05a79e695},
intrahash = {a472c93991deae5d681230bb2ebc5da9},
keywords = {Langlands},
note = {cite arxiv:2207.02955},
timestamp = {2022-07-09T08:22:11.000+0200},
title = {Non-vanishing of geometric Whittaker coefficients for reductive groups},
url = {http://arxiv.org/abs/2207.02955},
year = 2022
}