Let G'<G be an embedding of semisimple complex Lie groups. One of the fundamental problems in representation theory is about the description of the space of G'-invariants in a simple G-module, and the variation the dimension of this space as function of the highest weight. The Borel-Weil theorem allows to realize all irreducible G-modules as spaces of section of line bundles on the flag variety X=G/B. Then one may apply Hilbert-Mumford theory for the G'-action on X to study the spaces of invariants.   In a joint work with H. Seppänen, we aim at an explicit description of the G'-unstable locus for any line bundle on X. We have achieved this for certain classes of subgroups. This yields alternative proofs of some know results, as well as a description the GIT-classes of line bundles and some properties of the GIT-quotients. We show that many GIT-quotients are Mori dream spaces. We deduce results on the asymptotic growth of multiplicities of invariants in terms of a global Okounkov body for a suitable quotient.