On the classification of polar representations
On the classification of polar representations
复制标题
DOI：
10.1007/pl00004763
复制
发表时间：
1999-11
影响因子：
0.8
通讯作者：
J. Eschenburg;E. Heintze
中科院分区：
文献类型：
--
作者：
J. Eschenburg;E. Heintze
设V是欧几里德向量空间，K∧O (V)是紧子群。如果说K在V上的表示是极的，如果存在一个线性子空间Σ它满足每一个轨道并且在每一个交点垂直满足它。例如所谓的s表示，即对称空间的各向同性表示。Dadok （[D]）几乎明确地分类了所有不可约的极表示（参见定理9和随后在第129页的注释，以及[D]中第133、134和136页的列表）。由于他的分类，他得到任何极表示都是轨道等效于s表示，即在向量空间的等距识别之后，它具有与合适的s表示相同的轨道（参见[D]， Prop. 6）。我们称之为他的主要结果。Dadok对分类的证明包括将所有紧李群的所有不可约表示的集合巧妙地约简为有限多种情况（其中许多情况留给读者）。在[EH]中，我们给出了秩（即主轨道的余维数）不为2的主要结果的无分类概念证明。事实上，对于V上的任何极坐标表示，与ρ轨道相同的O (V)的最大子群是s表示。本文的目的是从主要结果推导出完整的分类。更确切地说，对于任何不可约对称空间S= G/K，我们将在S的各向同性表示下确定K的所有（连通）子群K/与K具有相同的轨道。这一证明的主要思想与以下观察相一致：主K轨道是等参子流形，由K和K/ K作用于其上的曲率球分叶
Let V be a euclidean vector space and K⊂ O (V) a compact subgroup. The representation of K on V is said to be polar, if there exists a linear subspace Σ which meets every orbit and meets it perpendicularly at every point of intersection. Examples are the so called s-representations, ie the isotropy representations of symmetric spaces. Dadok ([D]) has classified almost explicitly all irreducible polar representations (cf. Theorem 9 and the subsequent remark on p. 129, together with the lists on pp. 133, 134 and 136 in [D]). As a consequence of his classification he gets that any polar representation is orbit equivalent to an s-representation, ie it has the same orbits as a suitable s-representation after an isometric identification of the vector spaces (cf.[D], Prop. 6). We will call this his main result. Dadok's proof for the classification consists in an ingeneous reduction of the set of all irreducible representations of all compact Lie groups to finitely many cases (many of which are left to the reader). In [EH] we gave a classification free conceptual proof of the main result if the rank (ie the codimension of the principal orbits) is not two. In fact it turns out that for any polar representation ρ on V, the maximal subgroup of O (V) with the same orbits as ρ is an s-representation. The purpose of the present paper is to derive the complete classification from the main result. More precisely, for any irreducible symmetric space S= G/K, we will determine all (connected) subgroups K/ of K having the same orbits as K undertheisotropyrepresentationofS. Themainideaofourproofconsistsin the observation that the principal K-orbits are isoparametric submanifolds which are foliated by curvature spheres on which K and also K/⊂ K act