Hodge Structures Attached to Geometric Automorphic Forms
Hodge Structures Attached to Geometric Automorphic Forms
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附加到几何自守形式的 Hodge 结构
DOI：
10.2969/aspm/00710223
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
T. Oda
中科院分区：
文献类型：
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作者：
T. Oda
这篇简短的注记介绍了实半单李群的离散级数的表示，以及离散子群的上同调群的Hodge理论。本文中的所有材料都在文献中找到，只是对证明做了一些微小的改动。这几年来，我一直希望有人能写一篇文章，其中包含了数学家们一直想知道的关于离散级数的一切……但却羞于询问。因此，这篇文章部分是为我自己写的。对离散级数的一切讨论都超出了我的能力范围，因为我在表象理论方面几乎没有经验。但在第一章中，我试图解释离散级数的基本结果：它们的定义、特征、实现以及K型定理(即Blattner猜想)。没有给出证据。我参考Harish-Chandra的论文和Warner的教科书[41]，以获得关于么正表示和离散级数特征的基本事实的证明。因为Narasimhan-Okamoto[31]的实现最适合我们的目的，所以我将在1.3节中讨论它。我还提到了Schmid[38]的实现，它适用于更一般的李群。省略了通过旋量实现单身性[35]。施密德[40]和赫克特-施密德[19]完成了布拉特纳猜想的证明。我们只记得它在第1.4节中的声明。在第二章中，我讨论了自同构上同调群，或自同构调和形式，即生成离散级数表示的自同构形式。本章最重要的结果是Parthasarathy[34]的消失定理(2.3.1)，它是对[15]、[31]、[38]中消失结果的最大改进。它是系数在某些全纯向量丛中的上同调群在有界对称域的算术商上的纯维。标题中的几何自同构形式是
This short note is an introduction to representations of discrete series of real semisimple Lie groups, and to the Hodge theory of cohomology groups of discrete subgroups. All the materials in this paper are found in the literature except for minor changes of proofs. For these several years, I have been hoping that someone would write an article which contains "everything that number theorists have always wanted to know about discrete series . . . but were ashamed to ask". So this is written partly for myself. To discuss everything on discrete series is out of my power, who have little experience in the representation theory. But in Chapter 1, I attempt to explain the basic results for discrete series: their definition, characters, realizations, and the K-type theorem (i.e. Blattner's conjecture). The proofs are not given. I refer to the papers of Harish-Chandra and the textbook of Warner [41] for the proofs of the fundamental facts on unitary representations and characters of discrete series. Because the realization of Narasimhan-Okamoto [31] is most suitable for our purpose, I discuss it in Section 1.3. Also I refer to the realization of Schmid [38] which is applicable to more general Lie groups. The realization of Parthasarathy [35] by spinors is omitted. The proof of Blattner's conjecture is completed by Schmid [40] and Hecht-Schmid [19]. We recall only its statement in Section 1.4. In Chapter 2, I discuss automorphic cohomology groups, or automorphic harmonic forms, i.e. automorphic forms which generate representations of discrete series. The most important result in this chapter is the vanishing theorem (2.3.1) of Parthasarathy [34], which is the sharpest improvement of the vanishing results in [15], [31], [38]. It is a puredimensionality of cohomology groups with coefficients in certain holomorphic vector bundles on arithmetic quotients of bounded symmetric domains. Geometric automorphic forms in the title are elements of the