A Cuspidality Criterion for the Functorial Product on GL(2) times GL (3), with a Cohomological Application

A strong impetus for this paper came, at least for the first author, from a question of Avner Ash, asking whether one can construct non-self-dual, nonmonomial cuspidal cohomology classes for suitable congruence subgroups Γ of SL(n,Z), say for n = 6. Such a construction, in special examples, has been known for some time for n = 3 (see [2, 34, 35, 36]); it is of course not possible for n = 2. One can without trouble construct non-selfdual, monomial classes for any n = 2m with m ≥ 2, not just for constant coefficients (see the appendix, Theorem A.1). In the appendix, we also construct nonmonomial, non-selfdual classes for n = 4 using the automorphic induction to Q of suitable Hecke character twists of non-CM cusp forms of “weight 2” over imaginary quadratic fields, but they admit quadratic self-twists and are hence imprimitive. The tack pursued in the main body of this paper, and which is the natural thing to do, is to take a non-self-dual (nonmonomial) n = 3 example π, and take its functorial product with a cuspidal π ′ on GL(2)/Q associated to a holomorphic newform of weight 4 for a congruence subgroup of SL(2,Z). The resulting (cohomological) n = 6 example can be shown to be non-self-dual for suitable π ′. (This should be the case for all π ′, but we cannot prove this with the current technology—see Remark 5.2.) Given that, the main problem is that it is not easy to show that such an automorphic tensor product Π := π π ′, whose modularity was established in the recent deep work of Kim and Shahidi (see [20]), is cuspidal. This has led us to prove