2010/01/27 by Kiumars Kaveh, Kaveh, Kiumars, Askold G. Khovanskii +1
Mathematics · #14L30 #52A39 #53D20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.AG #math.RT #math.SG #msc:14L30 #msc:52A39 #msc:53D20
paper · pdf · doi:10.48550/arxiv.1001.4830
23 pages. Revised in several places and made considerably shorter. Final version, to appear in Moscow Mathematical Journal volume in honor of V. I. Arnold
arxiv created 2012/03/28 · arxiv updated 2012/03/30
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-Kazarnowski 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 our ealier work arXiv:0904.3350