论文标题
在Borel-Serre边界上翻译的平等分布
Equidistribution of translates of a homogeneous measure on the Borel--Serre boundary
论文作者
论文摘要
令G为在有理数上定义的半密布线性代数群，k为其实际点的最大紧凑型亚组，γ是算术晶格。可以将Q定义的每个子组H的γ\ G上的概率度量μ（h）关联，而没有非平凡的有理特征。当g从右侧作用于γ\ g时，我们可以通过将这些措施推向γ\ g/k的元素来推动这一措施，我们称它们称为同质。询问同质措施的可能弱限制是一个自然的问题。在非发散的情况下，这是由Eskin-Mozes-Shah回答的。在发散的情况下，daw-gorodnik- ullmo证明了对真实独立物生成的h的γ\ g/k的一些非平地压缩中的精制版本。在本文中，我们以他们的工作为基础，并将其推广到一般H的情况下，没有非平凡的理性字符。我们的结果依赖于（1）SL_N上的非发散标准，该标准是由数字的几何形状和kleinbock-margulis定理证明的； （2）与几何不变理论和还原理论证明的与不同群体相关的部分borel-serre压缩之间的关系。
Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and Γ be an arithmetic lattice. One can associate a probability measure μ(H) on Γ\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on Γ\G from the right, we can push-forward this measure by elements from G. By pushing down these measures to Γ\G/K, we call them homogeneous. It is a natural question to ask what are the possible weak-* limits of homogeneous measures. In the non-divergent case this has been answered by Eskin--Mozes--Shah. In the divergent case Daw--Gorodnik--Ullmo prove a refined version in some non-trivial compactifications of Γ\G/K for H generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general H with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on SL_n proved by geometry of numbers and a theorem of Kleinbock--Margulis; (2) relations between partial Borel--Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory.
