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Automorphic Gluing

We prove a gluing theorem on the automorphic side of the geometric Langlands correspondence: roughly speaking, we show that the difference between $\mathrm{DMod}(\mathrm{Bun}_G)$ and its full subcategory $\mathrm{DMod}(\mathrm{Bun}_G)^\mathrm{temp}$ of tempered objects is compensated by the categories $\mathrm{DMod}(\mathrm{Bun}_M)^\mathrm{temp}$ for all standard Levi subgroups $M \subsetneq G$. This theorem is designed to match exactly with the spectral gluing theorem, an analogous result occurring on the other side of the geometric Langlands conjecture. Along the way, we state and prove several facts that might be of independent interest. For instance, for any parabolic $P \subseteq G$, we show that the functors $\mathrm{CT}_{P,*}:\mathrm{DMod}(\mathrm{Bun}_G) \to \mathrm{DMod}(\mathrm{Bun}_M)$ and $\mathrm{Eis}_{P,*}: \mathrm{DMod}(\mathrm{Bun}_M) \to \mathrm{DMod}(\mathrm{Bun}_G)$ preserve tempered objects, whereas the standard Eisenstein functor $\mathrm{Eis}_{P,!}$ preserves anti-tempered objects.