Non-vanishing of geometric Whittaker coefficients for reductive groups

TL;DR

This paper proves the non-vanishing of Whittaker coefficients for cuspidal automorphic D-modules on reductive groups, extending geometric Langlands results and establishing new exactness properties of functors.

Contribution

It generalizes known results from GL_n to all reductive groups and introduces microlocal methods to interpret Whittaker coefficients.

Findings

Whittaker coefficients are non-vanishing for cuspidal automorphic D-modules.

Hecke functors are t-exact on tempered D-modules.

Whittaker coefficient functors are t-exact for sheaves with nilpotent singular support.

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.