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.