  This paper was motivated by a question of Avner Ash, asking if it is possible
to construct non-selfdual, non-monomial, cuspidal cohomology classes for
suitable congruence subgroups \Gamma of SL(n,\Z). Such a construction, in
special examples, has been known for some time for n=3; it is of course
impossible for n=2. We show in this paper the existence of many such examples
for n=6, which are primitive, by making use of the functorial product on GL(2)
x GL(3), which was recently shown to be automorphic by Kim and Shahidi. We
establish a general cuspidality criterion for this product, which is essential
to the construction. We also show that there exist non-selfdual, monomial
(cuspidal) classes for any n=2m > 3, and non-selfdual, non-monomial (but
imprimitive) classes for n=4.
