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We prove that cuspidal automorphic $D$-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from $G{L}_n$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 $G{L}_n$. 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 ${\mathrm{Bun}}_G$. 

Abstract Arinkin and Gaitsgory defined a category oftempered‐modules on that is conjecturally equivalent to the category of quasi‐coherent (not ind‐coherent!) sheaves on . However, their definition depends on the auxiliary data of a point of the curve; they conjectured that their definition is independent of this choice. Beraldo has outlined a proof of this conjecture that depends on some technology that is not currently available. Here we provide a short, unconditional proof of the Arinkin–Gaitsgory conjecture.