The Volume of the Fundamental Domain for Some Arithmetical Subgroups of Chevalley Groups
The Volume of the Fundamental Domain for Some Arithmetical Subgroups of Chevalley Groups
复制标题
Chevalley群的一些算术子群的基本域的体积
DOI：
10.1090/pspum/009/0213362
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
R. Langlands
中科院分区：
文献类型：
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作者：
R. Langlands
通常，存在与gQ相关的VC = VQ⊗QC的线性变换的连通代数群G。如果H是VQ中的某个格，满足(i) M =∑λ∈L M∩V (λ), (ii) （X α/n!）对于所有α，令GZ = {g∈g | gM = M}。设ω为最高次GR上的左不变形式，在(∧i=1H1)∧(∧α>0Xα)上取±1的值，设[dg]为与ω相关的哈尔测度。我们现在的目的是展示以下内容。若ξ（·）为Riemann zeta函数，Πi=1（ti + 1）为GC的poincar<e:1>多项式，c为GC的基群阶数，则为∫
As usual, there is associated to gQ a connected algebraic group G of linear transformations of VC = VQ ⊗ QC. If H is some lattice in VQ satisfying (i) M = ∑ λ∈L M ∩ V (λ), (ii) (X α/n!)M ⊆ M for all α, then we let GZ = {g ∈ G | gM = M}. Let ω be a left invariant form on GR of highest degree which takes the value ±1 on (∧i=1H1) ∧ (∧α>0Xα) and let [dg] be the Haar measure associated to ω. Our purpose now is to show the following. If ξ(·) is the Riemann zeta function, Πi=1(ti + 1) is the Poincaré polynomial of GC, and c is the order of the fundamental group of GC then ∫