Let P be a maximal parabolic with Levi M in a semisimple group G over the rationals. Assume P remains maximal over the real numbers, and that both M and G have discrete series. Under certain assumptions, the automorphic representations of G generated by residual Eisenstein series coming from M will have (g,K)-cohomology in degrees one below and one above middle; these assumptions include that the inducing cuspidal representation of M is discrete series at infinity (and K is connected), and they require a twist by a specific power of the modulus character. The class below middle degree will then contribute to the full automorphic cohomology of G.

In this talk, I will discuss a new result that shows that, despite this, the class above middle degree always vanishes in the full automorphic cohomology of G. To prove this, we first examine in detail the structure of the associated induced representation at infinity, giving a description of its constituents and proving a theorem about the order of vanishing of intertwining operators on this representation. Using this archimedean information, we then construct an explicit primitive of the residual class that appears above middle degree, thereby expressing that class as a coboundary.