%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 }