Eigenspaces of invariant differential operators on an affine symmetric space
Eigenspaces of invariant differential operators on an affine symmetric space
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DOI：
10.1007/bf01389818
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发表时间：
1980-02
影响因子：
3.1
通讯作者：
T. Oshima;J. Sekiguchi
中科院分区：
文献类型：
--
作者：
T. Oshima;J. Sekiguchi
与黎曼对称空间的分析相比，仿射对称空间的分析在数学和理论物理的许多领域中都很重要，但目前还没有得到很好的研究。对称空间的概念最初是由Caftan （cf. Nomizu[29]）引入的；Berger[1]完成了不可约仿射对称空间的分类。在非紧型黎曼对称空间G/K上，所有不变微分算子的任何联合特征函数都可以用泊松积分给出。更确切地说，泊松变换是G的两个表示空间之间的G同构，即G/K上所有不变微分算子的联合特征空间与G/P上某线束的超函数值截面空间之间的G同构，其中P是G的最小抛物子群。这是Helgason[14]猜想，并由Kashiwara et al.[17]证明。这一证据依赖于Kashiwara和Oshima之前的一项研究。鉴于上述成功，我们提出了以下问题：仿射对称空间G/H上不变微分算子的联合特征-超函数是否可以用G/H“边界”上的超函数的“泊松积分”来表示？本文的目的是证明对于一类仿射对称空间，这个答案是相当肯定的。因此，我们的工作重点是消除h的非紧性所造成的困难。b[31]中已经公布了部分结果。本文组织如下。首先，我们引入了关于有限中心实数半单李群G的一些标准符号。在
In contrast to the analysis on a Riemannian symmetric space, analysis on an affine symmetric space has not yet been well studied in spite of its importance in many fields of mathematics and theoretical physics. The notion of symmetric spaces is originally introduced by Caftan (cf. Nomizu [29]); and the classification of irreducible affine symmetric spaces is accomplished by Berger [1]. It should be recalled that on a Riemannian symmetric space G/K of noncompact type any joint eigenfunction of all invariant differential operators can be given by the Poisson integral. More precisely, the Poisson transformation is a G-isomorphism between the two representation spaces of G, namely, between the joint-eigenspace of all invariant differential operators on G/K and the space of hyperfunction-valued sections of a certain line bundle over G/P, where P is a minimal parabolic subgroup of G. This was conjectured by Helgason [14] and proved by Kashiwara et al.[17]. The very proof depends on a previous study by Kashiwara and Oshima [18]. In view of the above success, we raise the following question: Can any joint eigen-hyperfunction of invariant differential operators on an affine symmetric space G/H be represented by the" Poisson integral" of hyperfunctions on the" boundary" of G/H?The purpose of the present paper is to show that the answer is quite affirmative for a certain class of affine symmetric spaces. Our effort is accordingly focused on the removal of difficulties caused by the non-compactness of H. A partial result is already announced in [31]. This paper is organized as follows. First we introduce some standard notation concerning a real semisimple Lie group G with finite center. In