齐次簇上格轨道的分布
Distribution of lattice orbits on homogeneous varieties
摘要：
给定局部紧群G中的格Γ和G的闭子群H，在齐性空间V=H \G上有自然作用Γ。对于有限子集的递增族{Γ_T：T>0}，稠密轨道vΓ，v \in V，以及V上的紧支撑函数V，我们考虑和S_{ϕ,v}（T）= \sum_{γ\in Γ_T} V γ.理解S_{ϕ,v}（T）的渐近行为是一个微妙的问题，它只考虑了H，G和{Γ_T}的某些非常特殊的选择。我们开发了一个一般的抽象方法的问题，并将其应用到的情况下，当G是一个李群和H或G是半单的。当G是一组具有范数的矩阵时，我们有S_{ϕ,v}（T）\sim \int_{G_T} n（vg）dg，其中G_T={g \in G：||g||且T_T = G_T \cap T。我们还证明了S_{ϕ,v}（T）的渐近性由\int_V dν决定，其中ν是依赖于v和范数的显式极限密度.
英文摘要：
Given a lattice Γin a locally compact group G and a closed subgroup H of G, one has a natural action of Γon the homogeneous space V=H\G. For an increasing family of finite subsets {Γ_T: T>0}, a dense orbit vΓ, v\in V, and compactly supported function ϕon V, we consider the sums S_{ϕ,v}(T)=\sum_{γ\in Γ_T} ϕ(v γ). Understanding the asymptotic behavior of S_{ϕ,v}(T) is a delicate problem which has only been considered for certain very special choices of H, G and {Γ_T}. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S_{ϕ,v}(T) \sim \int_{G_T} ϕ(vg) dg, where G_T={g\in G:||g||<T} and Γ_T = G_T \cap Γ. We also show that the asymptotics of S_{ϕ,v}(T) is governed by \int_V ϕdν, where νis an explicit limiting density depending on the choice of v and the norm.
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