论文标题
$ \ mathrm {sl} _2（\ mathbb {r}）$的固定度量
Stationary measures for $\mathrm{SL}_2(\mathbb{R})$-actions on homogeneous bundles over flag varieties
论文作者
论文摘要
让$ g $成为一个真正的半圣谎言组，具有有限的中心，没有紧凑的因素，抛物线子组的$ q <g $ $ x $ a $ g $的均匀空间承认对旗帜的$ g/q $的均等投影，由纤维带有纤维的纤维，由$ q $ $ q $的晶状体的晶状体量。给定概率度量$μ$，Zariski浓密的副本为$ H = \ Mathrm {sl} _2（\ Mathbb {r}）$在$ G $中，我们对$ $ $ $ $ stationary的可能性概率指标的描述，并在$ x $上进行，并验证了相应的均衡性结果。与对应于$ g = q $相对应的Benoist-Quint的结果相反，$ G = Q $的类型$μ$承认，相对于$ Q $，$ H $的位置很大。我们描述了可能的情况，并使用Eskin-Mirzakhani和Eskin-Lindenstrauss的作品中的思想来描述其中的所有案例。
Let $G$ be a real semisimple Lie group with finite centre and without compact factors, $Q<G$ a parabolic subgroup and $X$ a homogeneous space of $G$ admitting an equivariant projection on the flag variety $G/Q$ with fibres given by copies of lattice quotients of a semisimple factor of $Q$. Given a probability measure $μ$, Zariski-dense in a copy of $H=\mathrm{SL}_2(\mathbb{R})$ in $G$, we give a description of $μ$-stationary probability measures on $X$ and prove corresponding equidistribution results. Contrary to the results of Benoist-Quint corresponding to the case $G=Q$, the type of stationary measures that $μ$ admits depends strongly on the position of $H$ relative to $Q$. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin-Mirzakhani and Eskin-Lindenstrauss.
