Invariant measures of horospherical flows on noncompact homogeneous spaces
Invariant measures of horospherical flows on noncompact homogeneous spaces
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非紧均匀空间上的星球流的不变测度
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发表时间：
1978
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通讯作者：
S. Dani
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作者：
S. Dani
Veech在[18]中证明了G/F上的“horosphere flow”是唯一遍历的，这里G是一个没有非平凡紧因子的半单解析群，F是一个离散的余紧子群;也就是说，G/F上存在一个唯一的Borel概率测度，它在这个流下是不变的。另一方面，在半单群G中存在一类广泛的离散子群F，使得G/F容许G-不变概率测度，但非紧。整幺模矩阵的离散子群SL（2，Z）在所有幺模矩阵的群SL（2，IR）中，构成了上述现象的最简单的例子。然而，事实证明，非紧齐次空间上的准球面流不可能是唯一遍历的。这是因为存在包含次球面子群的真闭子群H，其闭轨道允许有限的H-不变测度。本文的目的是断言，对于包括上面引用的例子在内的某一类准球面流，所有各态历经不变测度都以上述方式出现。我们还记得遍历测度的集合也完全决定了所有有限不变测度的集合。实际上，我们并不局限于没有非平凡紧因子的半单群类，而是考虑任何约化解析群T。除了更一般之外，这在某些论证中也是方便的。称T中的子群U是次球面的，如果存在泰特使得
In [18] Veech proves that the "horospherical flow" on G/F where G is a semisimple analytic group with no non-trivial compact factor and F is a discrete co-compact subgroup, is uniquely ergodic; that is, there is a unique Borel probability measure on G/F which is invariant under the flow. On the other hand there exists a wide class of discrete subgroups F in semisimple groups G such that G/F admits a G-invariant probability measure, but is noncompact. The discrete subgroup SL(2,Z) of integral unimodular matrices in SL(2, IR) the group of all unimodular matrices, constitutes the simplest example of the above phenomenon. It turns out however, that the horospherical flow on a non-compact homogeneous space cannot be uniquely ergodic. This is because there exist proper closed subgroups H containing horospherical subgroups, having closed orbits which admit a finite H-invariant measure. The objective of the present paper is to assert that for a certain class of horospherical flows including the example cited above all the ergodic invariant measures arise in the above manner. We recall that the set of ergodic measures also determines completely the set of all finite invariant measures. Actually rather than restricting to the class of semisimple groups with no non-trivial compact factor we consider any reductive analytic group T. Apart from being more general this is also convenient in certain arguments. A subgroup U in T is said to be horospherical if there exists teT such that