About some faces of the generalized Littlewood-Richardson cone

Let G be a connected reductive algebraic group and H be a reductive closed and connected subgroup of G both defined on an algebraically closed field of characteristic zero. We consider the set C of the couple (x,y) of the dominant weights such that the irreducible H-module of highest weight x is a submodule of the G-module corresponding to y. This set identifies canonically with a semi-group finitely generated in a rational vector space. We consider the generalized Littlewood-Richardson cone C' generated by C. This cone is polyhedral. For any (x',y') in C', x' is dominant, and so, satisfies a finite number (rank of H) of linear inequalities. We ask if these inequalities induce faces of codimension one of C'. By Geometric Invariant Theory methods, we give geometric criteria that imply an affirmative answer. We also apply our results to several classical example in representation theory.