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Explore a lecture on the combinatorial description of GIT-classes and special properties of the GIT-fan in relation to branching laws for reductive groups. Delve into the translation of H-invariant vectors in finite dimensional representations of G into Geometric Invariant Theory for the H-action on the complete flag variety G/B. Examine the approach initiated by Heckman and developed by Berenstein-Sjamaar, Belkale-Kumar, and Ressayre, leading to a description of the H-ample cone. Investigate how representation theoretic information on H G can be encoded into properties of toric varieties associated with the GIT-fan and relevant lattices, based partly on joint work with Seppaenen.
