We consider the action of a connected reductive algebraic group G on the graded algebra A of sections of a line bundle on a projective variety X. The asymptotic of multiplicities of irreducible representations appearing in A is related to the Duistermaat-Heckman function and Riemann-Roch theorem for multiplicities due to Guillemin-Sternberg, Meinrenken and others.
For a given representation \lambda let m_{k, \lambda} denote the multiplicity of \lambda appearing in the k-th degree piece A_k . We describe the asymptotic behavior of m_{k, \lambda} as k goes to infinity. Our methods have elementary convex geometric nature and use the theory of Newton-Okounkov bodies. This work recovers and extends some previous results and of Brion and Paoletti who obtain similar results using Reimann-Roch theorem for multiplicities.This is a joint work with Takuya Murata.