This article is cited in 17 papers

Convex bodies associated to actions of reductive groups

Kiumars Kaveh^a, Askold G. Khovanskii<sup>bcd

a</sup> Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
^b Mathematics, University of Toronto, Toronto, Canada
^c Moscow Independent University
^d Institute for Systems Analysis, Russian Academy of Sciences

Abstract: 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].

Key words and phrases: Reductive group action, multiplicity of a representation, Duistermaat–Heckman measure, moment map, graded $G$-algebra, $G$-line bundle, volume of a line bundle, semigroup of integral points, convex body, mixed volume, Brunn–Minkowski inequality.

MSC: Primary 14L30, 53D20; Secondary 52A39

Received: January 17, 2011; in revised form January 14, 2012

Language: English

DOI: 10.17323/1609-4514-2012-12-2-369-396

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