Generalizations of the PRV conjecture, II

Let G⊂Gˆ be two complex connected reductive groups. We deal with the problem of finding sub-G  -modules of a given irreducible Gˆ-module. In the case where G   is diagonally embedded in Gˆ=G×G, S. Kumar and O. Mathieu found some of them, proving the PRV conjecture. Recently, the authors generalized the PRV conjecture on the one hand to the case where Gˆ/G is spherical of minimal rank, and on the other hand giving more sub-G  -modules in the classical case G⊂G×GG⊂G×G. In this paper, these two recent generalizations are combined in a more general result.