Convex bodies associated to actions of reductive groups
Convex bodies associated to actions of reductive groups
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与还原基团作用相关的凸体
DOI：
10.17323/1609-4514-2012-12-2-369-396
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
A. Khovanskii
中科院分区：
文献类型：
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作者：
Kiumars Kaveh;A. Khovanskii
我们将凸体与一类广义的分次G-代数联系起来，其中G是一个连通的约化群。这些凸体给出了关于希尔伯特函数的信息以及出现在分次代数中的不可约表示的多重性。本文将Duistermaat-Heckman测度的概念推广到分次G-代数上，证明了该测度的一个Fujita型逼近定理和一个Brunn-Minkowski不等式。这特别适用于任意G-线丛给出了线丛体积理论的等变版本。我们推广的Brion-Kazarnowski公式的程度，一个球形的品种，以任意的G-品种。我们的方法遵循了A. Okounkov.我们使用在我们早期的工作arXiv：0904.3350中发展的整点半群和Newton-Okounkov体的渐近理论
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