On the Eisenstein functoriality in cohomology for maximal parabolic subgroups

In his paper, ’On torsion in the cohomology of locally symmetric varieties’, Peter Scholze has introduced a new, purely topological method to construct classes on arithmetic quotients spaces reductive groups over $$\mathbb {Q}$$ originating from similar Levi subgroups maximal parabolic subgroups. We extend this construction beyond cases he considers, and, complex case, local systems.