ON REPRESENTATIONS AND COMPACTIFICATIONS OF SYMMETRIC RIEMANNIAN SPACES
ON REPRESENTATIONS AND COMPACTIFICATIONS OF SYMMETRIC RIEMANNIAN SPACES
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对称黎曼空间的表示和紧化
DOI：
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发表时间：
1960
期刊：
影响因子：
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通讯作者：
I. Satake
中科院分区：
文献类型：
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作者：
I. Satake
在最近对自守函数的研究[2]，[4]，[7]中，有必要考虑对称有界域（或更一般地，对称黎曼空间）的边界，并明确考虑这些边界上函数的行为。现在，“边界”的概念自然地以空间的“紧化”为前提。本文给出了利用对称黎曼空间的某些等距将对称黎曼空间紧化为所有正定厄米特矩阵空间的一般方法。设S是非紧型对称黎曼空间，即，S = GIK，G是一个中心有限的（连通）半单李群，其单因子都是非紧的，G的Ka极大紧子群.设g，f分别是G和K的李代数.则对于n维（复）向量空间V中g的任意忠实不可约表示p，我们可以在V中取一个合适的基，使得p（zX）=p（X）对于Xegg，z表示g的自同构，由S围绕x 0 = K的对称性定义;然后我们有G到PSL（n，使得对应于k ∈ K的矩阵p（k）是酉矩阵。然后使x = gK e S对应于Hermitian矩阵p（g）p（g），我们得到S到空间的一个注入，也记为p？所有n次行列式为1的正定埃尔米特矩阵的P1。这个映射p从S到91是一个等距（相对于一些不变的度量在S和？E1），并满足下列条件：（i）p（S）在？（ii）p（S）是不可约矩阵集，（iii）p（x 0）i_ n是n次单位矩阵。反过来，它可以很容易地证明，任何等距p从S到？满足这些条件的n是通过上述方法得到的;我们称这样的等距为S到9 nF的不可约表示。现在用P（iCn）表示与所有n次厄米特矩阵的（真实的）向量空间Mn相关联的（真实的）射影空间。当1正则嵌入P（3Cn）中时，P（S）可以看作是P（？）n）;
In the recent study of automorphic functions [2], [4], [7], it has become necessary to take boundaries of symmetric bounded domains (or more generally, of symmetric Riemannian spaces) into account and to consider explicitly the behavior of the functions on these boundaries. Now the notion of "boundary" presupposes naturally a "compactification" of the space. The purpose of this paper is to give a general method of compactifying symmetric Riemannian spaces by means of certain isometries of them into the space of all positive definite hermitian matrices. Let S be a symmetric Riemannian space of non-compact type, i.e., S = GIK, G being a (connected) semi-simple Lie group with finite center such that all its simple factors are non-compact and Ka maximal compact subgroup of G. Let g, f be the Lie algebras of G and K, respectively. Then for any faithful irreducible representation p of g in a (complex) vector space V of dimension n, we can take a suitable base in V such that p(zX) =p(X) for X e g, z denoting the automorphism of g defined by the symmetry of S around x0 = K; we have then the corresponding irreducible projective representation, denoted also by p, of G into PSL(n, C) such that the matrices p(k) corresponding to k e K are unitary. Then making x = gK e S correspond to the hermitian matrix p(g)p(g), we obtain an injection, denoted also by p, of S into the space ?P1 of all positive definite hermitian matrices of degree n and of determinant 1. This mapping p from S into 91 is an isometry (with respect to some invariant metrics in S and in ?E1) and satisfies the following conditions: (i ) p(S) is totally geodesic in ?Pl, (ii) p(S) is an irreducible set of matrices, (iii) p(x0) i_ n (the identity matrix of degree n). Conversely, it can be proved easily that any isometry p from S into ?n satisfying these conditions is obtained by the above method; we call such an isometry an irreducible representation of S into 9nF. Now denote by P(iCn) the (real) projective space associated with the (real) vector space Mn of all hermitian matrices of degree n. On1 being imbedded canonically in P(3Cn), p(S) can be considered as a subset of P(?n);