EISENSTEIN SERIES
EISENSTEIN SERIES
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爱森斯坦系列
DOI：
10.1090/coll/058/07
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发表时间：
2008
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
在这些讲座中，我想用一些证明的迹象来讨论爱森斯坦级数理论中的一些基本事实。虽然讨论可以更一般地进行，但在本研究所的上下文中，将定义在Q上的约化群GC的实点的群G的算术定义的子群Γ作为离散群是最方便的，其连通分量GQ没有有理性质。还必须假设GC的一个最大Q分裂环面的中心子满足GC的每一个分支。波雷尔的约化理论，经过微不足道的修改，适用于G；假设Γ有一个只有一个尖点的基本集是很方便的。确定了定义在Q上的极小抛物线子群P0-C和P0-C的最大q-分裂环面A0-C，从而定义了标准抛物线Q-子群。一个(标准)尖(尖)子群P是G 0 C的一个(标准)抛物线(极小抛物线)Q-子群PC在G中的正规化子.每个标准尖(尖)子群P都与AC的李代数A0 C的一个子空间AC相联系，这个子空间称为P的分裂分支.根据定义，P的秩等于它的维度。Ac上实点的集合a也称为P的分裂分量。P是乘积AMN，其中A是G的具有李代数a的解析子群，N是PC的幂等根中的实点集，M满足与G相同的条件。我们将M与N\MN联系起来。则Γ∩P⊆MN和Θ=Γ∩N\Γ∩MN是M的算术定义的子群。假设对于每个标准尖点子群P，它也有一个只有一个尖点的基本域。
In these lectures I want to discuss, with some indications of proofs, some of the elementary facts in the theory of Eisenstein series. Although the discussion can be carried out in more generality it is most convenient, in the context of this institute, to take for discrete group an arithmetically defined subgroup Γ of the group G of real points of a reductive group GC defined over Q whose connected component GQ has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of GC meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defined over Q and a maximal Q-split torus A 0 C of P 0 C so that the standard parabolic Q-subgroups are defined. A (standard) cuspidal (percuspidal) subgroup P is the normalizer in G of a (standard) parabolic (minimal parabolic) Q-subgroup PC of G 0 C. To each standard cuspidal subgroup P is associated a subspace AC of the Lie algebra a 0 C of AC; this subspace will be called the split component of P . By definition the rank of P is equal to its dimension. The set a of real points on aC will also be called the split component of P . P is a product AMN where A is the analytic subgroup of G with the Lie algebra a, N is the set of real points in the unipotent radical of PC, and M satisfies the same conditions as G. We identify M with N\MN . Then Γ ∩ P ⊆ MN and Θ = Γ ∩ N\Γ ∩MN is an arithmetically defined subgroup of M . Assume that for each standard cuspidal subgroup P it also has a fundamental domain with only one cusp.