Compactifications of Locally Symmetric Spaces
Compactifications of Locally Symmetric Spaces
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局部对称空间的紧化
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
L. Ji
中科院分区：
文献类型：
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作者：
A. Borel;L. Ji
设G是定义在Q上的连通半单线性代数群G的实迹，Γ⊂G(Q)是算术子群。然后，商Γ\G是一个自然齐次空间，它的一个极大紧子群K右边的商构成了一个局部对称空间Γ\G/K。本文构造了Γ\G的几个新的紧化，前两个与局部对称空间Γ\G/K的Borel-Serre紧化和约化Borel-Serre紧化有关，实际上，它们给出了这些已知紧化的另一种构造。更重要的是，Γ\G的紧化蕴含着Γ\G/K上齐次丛的紧化的推广，并且在非极大紧子群H下的这些紧化给出了周期域Γ\G/H的紧化。Γ\G的另一个紧化是通过嵌入到G的闭子群空间中得到的，它与自同构形式的常项密切相关，特别是与Eisenstein级数密切相关。
Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a natural homogeneous space, whose quotient on the right by a maximal compact subgroup K of G gives a locally symmetric space Γ\G/K. This paper constructs several new compactifications of Γ\G. The first two are related to the Borel-Serre compactification and the reductive Borel-Serre compactification of the locally symmetric space Γ\G/K; in fact, they give rise to alternative constructions of these known compactifications. More importantly, the compactifications of Γ\G imply extension to the compactifications of homogeneous bundles on Γ\G/K, and quotients of these compactifications under non-maximal compact subgroups H provide compactifications of period domains Γ\G/H in the theory of variation of Hodge structures. Another compactification of Γ\G is obtained via embedding into the space of closed subgroups of G and is closely related to the constant term of automorhpic forms, in particular Eisenstein series.