Global branching laws by global Okounkov bodies

Abstract

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