The Adjoint Representation and the Adjoint Action
The Adjoint Representation and the Adjoint Action
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伴随表示和伴随动作
DOI：
10.1007/978-3-662-05071-2_3
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
W. Mcgovern
中科院分区：
文献类型：
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作者：
W. Mcgovern
本文的目的是详细研究半单李群或代数群在其李代数上的伴随表示作用和在其自身上的伴随作用。我们将主要关注通过李代数中幂零元的轨道;这些轨道简称为幂零轨道。在过去的35年里，人们得到了许多关于这种轨道的深刻结果;我们将收集其中一些在表示论中有广泛应用的最重要的结果。我们将主要研究特征为零的代数闭域上的半单李代数及其伴随群，但我们将把我们所做的大部分工作扩展到实数或素特征的代数闭域上的半单李代数，以及半单代数群中的共轭类。我们将给出许多结果的详细证明，其中包括一些很难从文献中找到的结果。其他结果将通过合理完整的参考文献进行总结。这一处理是[CM 93]中更全面的版本;也与Humphreys的书[Hu 95]有一些重叠。在最后一章中，我们总结了一些最新的工作正在做的这一主题，并指出了当前的研究方向。
The purpose of this article is to study in detail the actions of a semisimple Lie or algebraic group on its Lie algebra by the adjoint representation and on itself by the adjoint action. We will focus primarily on orbits through nilpotent elements in the Lie algebra; these are called nilpotent orbits for short. Many deep results about such orbits have been obtained in the last thirty-five years; we will collect some of the most significant of these that have found wide application to representation theory. We will primarily work in the setting of a semisimple Lie algebra and its adjoint group over an algebraically closed field of characteristic zero, but we will extend much of what we do to semisimple Lie algebras over the reals or an algebraically closed field of prime characteristic, and to conjugacy classes in semisimple algebraic groups. We will give detailed proofs of many results, including some which are difficult to ferret out of the literature. Other results will be summarized with reasonably complete references. The treatment is a more comprehensive version of that in [CM93]; there is also some overlap with Humphreys’s book [Hu95]. In the last chapter we summarize some of the most recent work being done in this topic and indicate some directions of current research.