Decomposition of Symmetric Powers of Irreducible Representations of Semisimple Lie Algebras and the Brion Polytope

To any closed irreducible G-invariant cone in the space V of a finitedimensional representation of a semisimple Lie group there corresponds a convex polytope called the Brion polytope. This is closely connected with the action of the group G on the algebra of functions on the cone, and also with the moment map. In this paper we give a description of Brion polytopes for the spaces V themselves and for their nullcones.