A Classification of the Principal Nilpotent Pairs in Simple Lie Algebras and Related Problems
A Classification of the Principal Nilpotent Pairs in Simple Lie Algebras and Related Problems
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简单李代数中主幂零对的分类及相关问题
DOI：
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发表时间：
1999
期刊：
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通讯作者：
D. Panyushev
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文献类型：
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作者：
A. Elashvili;D. Panyushev
设g是特征为零的代数闭域K上的半单李代数，G是它的伴随群。主幂零对的概念是g中正则（=主）幂零元概念的双重对应。粗略地说，主幂零对e =（e1，e2）由g中的两个可交换元素组成，它们可以独立地收缩到原点，并且使得它们的同时中心化子具有最小可能维数，即rkg。这个定义和基本结果是由V.Ginzburg [3]提出的。他证明了主幂零对理论对B的著名结果进行了改进。Kostant对正规幂零元素在g和有有趣的应用表示论。特别地，他证明了主幂零对的G轨道的数目是有限的，并给出了g = sl（V）的分类。为了达到更大的普遍性，金斯堡还引入了更广泛的一类杰出的幂零对，并再次将它们归类为sl（V）。(The与幂零对相关的所有概念的精确定义见§1。）
Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in g. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rkg. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well‐known results by B. Kostant on regular nilpotent elements in g and has interesting applications to representation theory. In particular, he proved that the number of G‐orbits of principal nilpotent pairs is finite and gave a classification for g = sl(V). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for sl(V). (The precise definitions for all notions related to nilpotent pairs are found in §1.)