#### Vol. 306, No. 1, 2020

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Nonvanishing square-integrable automorphic cohomology classes: the case GL(2) over a central division algebra

### Joachim Schwermer

Vol. 306 (2020), No. 1, 321–355
##### Abstract

Let $k$ be a totally real algebraic number field, and let $D$ be a central division algebra of degree $d$ over $k$. The connected reductive algebraic $k$-group $GL\left(2,D\right)∕k$ has $k$-rank one; it is an inner form of the split $k$-group $GL\left(2d\right)∕k$. We construct automorphic representations $\pi$ of $GL\left(2d\right)∕k$ which occur nontrivially in the discrete spectrum of $GL\left(2d,k\right)$ and which have specific local components at archimedean as well as nonarchimedean places of $k$ so that there exist automorphic representations ${\pi }^{\prime }$ of $GL\left(2,D\right)\left({\mathbb{𝔸}}_{k}\right)$ with $\Xi \left({\pi }^{\prime }\right)=\pi$ under the Jacquet–Langlands correspondence. These requirements depend on the finite set ${V}_{D}$ of places of $k$ at which $D$ does not split, and on the quest to construct representations ${\pi }^{\prime }$ of $GL\left(2,D\right)\left({\mathbb{𝔸}}_{k}\right)$ which either represent cuspidal cohomology classes or give rise to square-integrable classes which are not cuspidal, that is, are eventually represented by a residue of an Eisenstein series. The demand for cohomological relevance gives strong constraints at the archimedean components.

##### Keywords
cohomology of arithmetic groups, square-integrable automorphic representations, general linear group over a central division algebra
##### Mathematical Subject Classification 2010
Primary: 11F70, 11F75
Secondary: 22E40, 22E55
##### Milestones
Accepted: 30 December 2019
Published: 14 June 2020
##### Authors
 Joachim Schwermer Faculty of Mathematics University of Vienna Vienna Austria Max Planck Institute for Mathematics Bonn Germany