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Abstract
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Let
be a totally real algebraic number field, and let
be a central division
algebra of degree
over
. The connected
reductive algebraic
-group
has
-rank one; it is an inner
form of the split
-group
. We construct automorphic
representations
of
which occur nontrivially
in the discrete spectrum of
and which have specific local components at archimedean as well as nonarchimedean places of
so that there exist
automorphic representations
of
with
under
the Jacquet–Langlands correspondence. These requirements depend on the finite set
of places of
at
which
does not split, and on the quest to construct representations
of
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.
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Keywords
cohomology of arithmetic groups, square-integrable
automorphic representations, general linear group over a
central division algebra
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Mathematical Subject Classification 2010
Primary: 11F70, 11F75
Secondary: 22E40, 22E55
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Milestones
Received: 19 October 2018
Accepted: 30 December 2019
Published: 14 June 2020
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