On the residual Eisenstein cohomology of unitary groups

We investigate the residual Eisenstein cohomology of an arbitrary unitary group $U(V)$ attached to an arbitrary quadratic extension of number fields $E\mathrm{/}F$. Our focus lies on the contribution of the maximal parabolic $F$-subgroups of $U(V)$, for which we identify the cohomologically relevant poles of Eisenstein series and prove that the resulting residues all survive as non-trivial classes in automorphic cohomology in an explicit degree. To illustrate the range of phenomena involved, we study in detail the case of a unitary group over $F=\mathbb{Q}(\sqrt[3]{2})$ of $F$-rank $3$, for which we explicitly construct cuspidal automorphic representations, which satisfy all the assumptions of our main theorem and hence explicitly construct non-zero residual Eisenstein cohomology classes for this unitary group. The methods used in this construction are paradigmatic however, i.e., generalize to other unitary groups over other ground fields $F$ by the use of base change.