Convex bodies associated to actions of reductive groups

We associate convex bodies to a wide class of graded $G$-algebras where $G$ is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in the graded algebra. We extend the notion of Duistermaat–Heckman measure to graded $G$-algebras and prove a Fujita type approximation theorem and a Brunn–Minkowski inequality for this measure. This in particular applies to arbitrary $G$-line bundles giving an equivariant version of the theory of volumes of line bundles. We generalize the Brion–Kazarnowskii formula for the degree of a spherical variety to arbitrary $G$-varieties. Our approach follows some of the previous works of A. Okounkov. We use the asymptotic theory of semigroups of integral points and Newton–Okounkov bodies developed in [15].