Hyperpolar Actions on Symmetric Spaces
Hyperpolar Actions on Symmetric Spaces
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对称空间上的超极作用
DOI：
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发表时间：
2001
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通讯作者：
G. Thorbergsson
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作者：
E. Heintze;R. Palais;G. Thorbergsson
截面在M中自动完全测地线，如果它在诱导度规中也是平坦的，那么作用称为超极。本文研究紧对称空间上的超极作用，证明了它们的一些结构定理和分类定理，并研究了它们与Hilbert空间上的超极作用和仿射Kac-Moody代数的对合的关系。超极作用理论的起源至少可以追溯到20世纪50年代，当时博特[B]和博特和萨缪尔森[BS]引入了与之密切相关的“变完全”作用的概念。我们将在下面简要回顾一下我们所知道的这门学科的历史，但首先我们回顾一下对称空间理论中的一些基本符号和术语，这将在整个论文中用到。设G是紧的，半简单的，连通的李群，G = TG是它的李代数。G的子群K称为G的对称子群（对（G， K）称为对称对），如果G存在一个对合σ，使得（Kσ）0∧K∧Kσ，其中Kσ是σ的不动点集，（Kσ）0是Kσ的连通分量。在这种情况下，由G的任意AdG不变内积导出一个G不变度量的G/K是一个对称空间。相应的正交分解G = K⊕P称为与σ相关的Cartan分解。它完全由σ决定（与G上的内积无关）；即K和P分别是dσe: G→G的+1和- 1特征空间。可得，K、P满足[K， K]、[K, P]、[P， P]等特征托架关系。1.1定义。如果（G， K）是对称对，且G = K⊕P为对应的正则Cartan分解，则K在P上的伴随表示称为s表示。当然，在等价的情况下，s表示可以用对称空间G/K在eK（它的名字由此而来）的各向同性表示来标识。
section is automatically totally geodesic in M, and if it is also flat in the induced metric then the action is called hyperpolar. In this paper we study hyperpolar actions on compact symmetric spaces, prove some structure and classification theorems for them, and study their relation to polar actions on Hilbert space and to involutions of affine Kac-Moody algebras. The origin of the theory of hyperpolar actions can be traced back at least to the 1950’s, with Bott [B] and Bott and Samelson’s [BS] introduction of the closely related concept of a “variationally complete” action. We will give a brief review below of what we know of the history of this subject, but first we recall some basic notation and terminology from the theory of symmetric spaces that will be needed throughout the paper. Let G be a compact, semi-simple, connected Lie group, and G = TG e its Lie algebra. A subgroup K of G is called a symmetric subgroup of G (and the pair (G, K) is called a symmetric pair) if there is an involution σ of G such that (Kσ)0 ⊂ K ⊂ Kσ, where Kσ is the fixed point set of σ and (Kσ)0 is the connected component of Kσ. In this case G/K equipped with a G-invariant metric induced from any AdG invariant inner product for G is a symmetric space. The corresponding orthogonal decomposition G = K⊕ P is called the Cartan decomposition associated to σ. It is completely determined by σ (independent of the inner product on G); namely K and P are respectively the +1 and −1 eigenspaces of dσe : G→G . It follows that K and P satisfy the characteristic bracket relations [K, K] ⊆K , [K, P] ⊆P , and [P, P] ⊆K . 1.1 Definition. If (G, K) is a symmetric pair and G = K⊕ P the correspond- “refsrep ing Cartan decomposition, then the Adjoint representation of K on P is called an s-representation. Of course, up to equivalence the s-representation can be identified with the isotropy representation of the symmetric space G/K at eK (from which it gets its name).