Rationality for isobaric automorphic representations: the CM-case

In this note we prove a simultaneous extension of the author's joint result with M. Harris for critical values of Rankin-Selberg L-functions $L(s,\Pi \times {\Pi }^{\text{'}})$ (Grobner and Harris in J Inst Math Jussieu 15:711-769, 2016, Thm. 3.9) to (i) general CM-fields F and (ii) cohomological automorphic representations ${\Pi }^{\text{'}}={\Pi }_1⊞\cdots ⊞{\Pi }_k$ which are the isobaric sum of unitary cuspidal automorphic representations ${\Pi }_i$ of general linear groups of arbitrary rank over F. In this sense, the main result of these notes, cf. Theorem 1.9, is a generalization, as well as a complement, of the main results in Raghuram (Forum Math 28:457-489, 2016; Int Math Res Not 2:334-372, 2010. https://doi.org/10.1093/imrn/rnp127), and Mahnkopf (J Inst Math Jussieu 4:553-637, 2005).