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
$GL\left(2,D\right)\u2215k$ has
$k$rank one; it is an inner
form of the split
$k$group
$GL\left(2d\right)\u2215k$. We construct automorphic
representations
$\pi $ of
$GL\left(2d\right)\u2215k$ 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
$GL\left(2,D\right)\left({\mathbb{\mathbb{A}}}_{k}\right)$ 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
$GL\left(2,D\right)\left({\mathbb{\mathbb{A}}}_{k}\right)$ which
either represent cuspidal cohomology classes or give rise to squareintegrable 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.
