An intrinsic analysis of unitarizable highest weight modules
An intrinsic analysis of unitarizable highest weight modules
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可统一化最高权重模块的内在分析
DOI：
10.1007/bf01444551
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发表时间：
1990
影响因子：
1.4
通讯作者：
A. Joseph
中科院分区：
文献类型：
--
作者：
T. Enright;A. Joseph
1.1.设g是复单李代数。本文研究了~的真实的形式go的可酉化最高权模。这些模块被分类在I-EHW]和[Ja]中。在这两种分类中，对于经典情形SU（p，q），Mp（2n，P，.）和SO*（2n）或Kashiwara和Vergne I-KV]的相应结果。对于其余的经典情形以及所有的例外情形，论证包括各种非常详细的关于约化和f-结构的逐案计算。在[P]中（参见[EHW，3.6 and 3.9]），对Casimir元素的仔细操作给出了一组范数不等式，我们称之为Parthasarathy不等式，并表明酉性等价于模中f-最高权重的这组范数不等式的有效性。[EHW]和[Ja]都使用~-结构对酉性的这种分析，来逐个考虑酉性的最后一点。在这里我们分析了一般情况下最高权模的~-结构，给出了酉性的必要条件。然后，我们使用相同的不等式来证明最后一个约化点以及所谓的Wallach集i-W1，W2]中的模的酉性。最后，我们利用Jakobsen的技巧，用WaUach集中的模在最后一个约化点张紧模，完成了酉性的证明。在目前的工作中，我们相信分析是相当新颖的，而基本的情况下，基本上是免费的。特别是，我们的论点并不诉诸于
1.1. Let g be a complex simple Lie algebra. In this article we are concerned with unitarizable highest weight modules for a real form go of~. These modules were classified in I-EHW] and [Ja]. In both of these classifications the arguments relied on the Howe theory of dual pairs for the classical cases SU (p, q), Mp (2n, P,.) and SO*(2n) or the corresponding results of Kashiwara and Vergne I-KV] for SU (p, q) and Mp (2n, R). For the remaining classical cases as well as all the exceptional cases the arguments included a variety of very detailed case by case calculations of reducibility and f-structure.1.2. In [P](see also [EHW, 3.6 and 3.9]) a careful manipulation of Casimir elements gives a set of norm inequalities, which we call the Parthasarathy inequalities, and shows that unitarity is equivalent to the validity of this set of norm inequalities for the f-highest weights in the module. Both [EHW] and [Ja] use this analysis of unitarity by~-structure for some case by case considerations of the last point of unitarity. Here we analyze the~-structure of highest weight modules in the general setting to give necessary conditions for unitarity. We then use the same inequalities to prove the unitarity of the last reduction point as well as the modules in what is called the Wallach set i-W1, W2]. Finally we complete the proof of unitarity by using Jakobsen's trick of tensoring the module at the last reduction point with the modules in the WaUach set. In the present work we believe the analysis to be quite novel, rather elementary and essentially case by case free. In particular our arguments do not appeal to