A vanishing theorem for residual Eisenstein cohomology

TL;DR

This paper investigates the residual Eisenstein cohomology of semisimple groups, establishing vanishing results for classes above middle degree and providing explicit cochains valued in Eisenstein series to understand their structure.

Contribution

It proves a vanishing theorem for residual Eisenstein cohomology classes above middle degree and constructs explicit cochains using regular Eisenstein series, detailing the archimedean component.

Findings

Residual classes above middle vanish in automorphic cohomology

Explicit cochains provide primitives for nontrivial classes

Archimedean component contains a sum of two discrete series

Abstract

We study the residual Eisenstein cohomology of semisimple groups in the context of maximal parabolic subgroups which remain maximal over $\mathbb{R}$. Under certain general hypotheses, we show that these residual representations are cohomological one degree below middle, and one above; however, the classes above middle vanish in the full automorphic cohomology. The proof of this vanishing finds an explicit cochain which provides a primitive to the image of a nontrivial class from the cohomology of such a residual representation. This cochain is valued in regular Eisenstein series. Along the way, we study in detail the archimedean component of the relevant induced representation. In particular, we prove that it has a subrepresentation which is the sum of two discrete series, whose Harish-Chandra parameters we describe, and that the intertwining operator vanishes to order exactly $1$ on that subrepresentation.