Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements
Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements
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单能元子群作用的齐次空间上的不变测度和轨道闭合
DOI：
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发表时间：
2000
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通讯作者：
N. Shah
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文献类型：
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作者：
N. Shah
本文将描述李群齐次空间上单幂流的有限遍历不变测度和轨道闭包的M. Ratner定理推广到由单幂元生成的子群的作用。更准确地说：设G是G的李群（不一定连通）和闭子群。设W是G的子群，使得AdG(W)包含在由AdG(W)的单幂元生成的子群的Zariski闭包（在Aut（Lie G）中）中。那么G/上的任何有限w不变w遍历测度都是齐次测度（即它被支持在保持该测度的子群的闭轨道上）。此外，如果G/具有有限体积（即具有有限G不变测度），则W在G/上的任何轨道的闭包是齐次集（即包含W的子群的有限体积闭包轨道）。如果W被任意子群替代，使得W/ W具有有限体积，上述两个结果都成立。
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and a closed subgroup of G. Let W be a subgroup of G such that AdG(W) is contained in the Zariski closure (in Aut(Lie G)) of the subgroup generated by the unipotent elements of AdG(W). Then any finite W-invariant W-ergodic measure on G/ is a homogeneous measure (i.e., it is supported on a closed orbit of a subgroup preserving the measure). Moreover, if G/ has finite volume (i.e., has a finite G-invariant measure), then the closure of any orbit of W on G/ is a homogeneous set (i.e., a finite volume closed orbit of a subgroup containing W). Both the above results hold if W is replaced by any subgroup � ⊂ W such that W/� has finite volume.