  By properly specializing the parameters irreducible modules of maximal
dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra
in type $A$ may be transformed into modules over Lusztig's infinitesimal
quantum algeba. Thus obtained modules have a simple socle and a simple head,
and share the same dimension as the infinitesimal Verma modules. Despite these
common features we find that they are never isomorphic to infinitesimal Verma
modules unless they are irreducible. The same carry over to the modular setup
for the special linear groups in positive characteristic.
