A cohomological injectivity result for the residual automorphic spectrum of $\mathrm{GL}_n$

Harald Grobner

Vol. 268 (2014), No. 1, 33–46

DOI: 10.2140/pjm.2014.268.33

Abstract

Let $\Pi$ be a cohomological residual automorphic representation of ${\text{ GL}}_n∕F$, for $F$ an arbitrary number field. Let $q_{min}$ be the lowest degree in which $\Pi$ has nonvanishing cohomology. We prove that the cohomology of $\Pi$ always injects into the cohomology of the corresponding locally symmetric space in degree $q_{min}$. This extends the well-known result of Borel for cuspidal automorphic representations to all square-integrable automorphic representations in this certain degree. Moreover, we thereby improve a result of Rohlfs and Speh and confirm an idea of Harder.

Keywords

automorphic cohomology, residual representation, general linear group

Mathematical Subject Classification 2010

Primary: 11F70, 11F75, 22E47

Secondary: 11F67

Milestones

Received: 20 December 2013

Revised: 4 March 2014

Accepted: 15 March 2014

Published: 21 May 2014