Originally, I wrote: Assuming that I understand the intent of the question properly, it is intended that psi be a Hecke character on the center of GL(2, adeles). The question seems to grant that we understand the situation with non-trivial Hecke characters subject only to the requirement that they be trivial on the "ray" ("the non-compact part" of the idele class group) denoted R-times-sub-plus. Then all other cases can be reduced to this by tensoring with characters of the form |det|^{it} for suitable real t. (Note that for L^2 to make sense the central character should be unitary.)

That is, nothing really new comes up.

Edit: Upon further consideration, I think that perhaps the question had some implicit questions in it, which can be answered:

The whole space of L^2 automorphic forms, even with trivial central character, includes many things that are not immediately classical holomorphic automorphic/modular forms, nor Maass/wave-forms. The (relatively easy) structure theory of representations of GL(2,R) or SL(2,R) shows that a Gamma-invariant K-finite Casimir-Laplacian eigenfunctions is either a holomorphic/anti-holomorphic automorphic form, a Maass waveform, or... significantly... a Lie algebra derivative of one of those.

The central character issue is somewhat secondary to this classification, which itself is not so hard.

In particular, perhaps we have been lucky that historical events presented us with "vectors" which generate all the relevant representations.