Okounkov bodies for ample line bundles with applications to multiplicities for group representations

Let $\mathcal{L}\to X$ be an ample line bundle over a complex normal projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$ofsubvarietiesforwhichtheassociatedOkounkovbodyfor$\mathscr{L}$isarationalpolytope\mathrm{.}Inthecasewhen$X$ is a homogeneous surface, and the pseudoeffective cone of $X$ is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone if the flag of subvarieties is suitably chosen. Finally, we provide an application to the asymptotic study of group representations.