  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.
