Geometric Eisenstein series I: finiteness theorems

We develop the theory of geometric Eisenstein series and constant term functors for $\mathrm{\ell }$-adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors, analogous to the usual finiteness properties of parabolic inductions and Jacquet modules. We also prove a geometric form of Bernstein’s second adjointness theorem, generalizing the classical result and its recent extension to more general coefficient rings proved in [Dat-Helm-Kurinczuk-Moss]. As applications, we decompose the category of sheaves on ${\mathrm{B}\mathrm{u}\mathrm{n}}_G$ into cuspidal and Eisenstein parts, and show that the gluing functors between strata of ${\mathrm{B}\mathrm{u}\mathrm{n}}_G$ are continuous in a very strong sense.