On arithmetic subgroups of a Q-rank 2 form of SU(2,2) and their automorphic cohomology

Abstract

The cohomology $H^{*}(\mathrm{\Gamma },E)$ of an arithmetic subgroup $\mathrm{\Gamma }$ of a connected reductive algebraic group $G$ defined over $\mathbf{Q}$ can be interpreted in terms of the automorphic spectrum of $\mathrm{\Gamma }$. In this frame there is a sum decomposition of the cohomology into the cuspidal cohomology ( i.e., classes represented by cuspidal automorphic forms for $G$) and the so called Eisenstein cohomology. The present paper deals with the case of a quasi split form $G$ of $\mathbf{Q}$-rank two of a unitary group of degree four. We describe in detail the Eisenstein series which give rise to non-trivial cohomology classes and the cuspidal automorphic forms for the Levi components of parabolic $\mathbf{Q}$-subgroups to which these classes are attached. Mainly the generic case will be treated, i.e., we essentially suppose that the coefficient system $E$ is regular.