  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.
