Parabolic geometric Eisenstein series and constant term functors

We prove a compatibility between parabolic restriction of Whittaker sheaves and restriction of representations under the geometric Casselman-Shalika equivalence. To do this, we establish various Hecke structures on geometric Eisenstein series functors, generalizing results of Braverman-Gaitsgory in the case of a principal parabolic. Moreover, we relate compactified and non-compactified geometric Eisenstein series functors via Koszul duality. We sketch a proof that the spectral-to-automorphic geometric Langlands functor commutes with constant term functors.