Constructing cohomological cuspidal representations - the case $GL(2)$ over a central division algebra

Let $D$ be a central division algebra of degree $d$ over a totally real algebraic number field $k$. The reductive $k$-group $GL(2, D)/k$ (of $k$-rank one) is an inner form of the split $k$-group $GL(2d)/k$.  We show the existence of cuspidal automorphic representations of $GL(2, D)/k$ which contribute non-trivially to its cuspidal cohomology. One family of these representations are  CAP representations (also called shadows of Eisenstein series), that is, they are nearly equivalent to representations occurring in the residual spectrum of the split $k$-group $GL(2d)/k$.