Global branching laws by global Okounkov bodies

Abstract

Let $G'$ be a complex semisimple group, and let $G \subseteq G'$ be a semisimple subgroup. We show that the branching cone of the pair $(G, G')$, which (asymptotically) parametrizes all pairs $(W, V)$ of irreducible finite-dimensional $G$-representations $W$ which occur as subrepresentations of a finite-dimensional irreducible $G'$-representation $V$, can be identified with the pseudo-effective cone, $\overline{\mbox{Eff}}(Y)$, of some GIT quotient $Y$ of the flag variety of the group $G \times G'$. Moreover, we prove that the quotient $Y$ is a Mori dream space. As a consequence, the global Okounkov body $Δ(Y)$ of $Y$, with respect to some admissible flag of subvarieties of $Y$, is fibred over the branching cone of $(G, G')$, and the fibre $Δ(Y)_{(W, V)}$ over a point $(W, V)$ carries information about (the asymptotics of) the multiplicity of $W$ in $V$. Using the global Okounkov body $Δ(Y)$, we easily derive a multi-dimensional generalization of Okounkov's result about the log-concavity of asymptotic multiplicities.