Symmetry in a class of elementary representations of a semisimple complex Lie group
Symmetry in a class of elementary representations of a semisimple complex Lie group
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半单复李群的一类初等表示中的对称性
DOI：
10.1007/bf01076083
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发表时间：
1967
影响因子：
0.4
通讯作者：
D. P. Zhelobenko
中科院分区：
文献类型：
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作者：
D. P. Zhelobenko
术语“初等表示”是与初等调和有关的，也就是圆和直线上的指数。半单复群G的初等表示被定义为由极大可解子群K的任意特征标0~(K)诱导的群的表示，并自然地在一类函数G/K中实现。在这方面，得到了特征标~(K)的一个子系统的Gel‘fand-Naimmark的么正表示。但在一般情况下，在G/K上的无穷可微函数类中定义一个初等表示e(~)是很自然的。可以把e(~)看作是Gel‘fand-Naimmark的“基本级数”的解析延续。然而，用群G的所有不可约有限维(实数)表示的级数代替基本级数也是可能的。直到最近，人们对任意(非么正)表示e(o~)的结构知之甚少。对这个问题的特别兴趣源于对群G[2-4，6，7，10，12]上的有限函数或快速递减函数的群环的研究。在这一点上，人们发现在初等表示类中存在双重对称性。一方面，在这一类中，Weyl群表现出自然的行为，其轨道对应于特征标类~，~(K)中的轨道，其变换归结为表示e(~)上的等价关系(对于“基本级数n的表示，这种对称性的存在在IM Gel‘land和MA Naimmark[5]的初始论文中已建立)；然而，在个别~奇异”点，这种关系退化。另一方面，从[2，3，6，7，10]，在奇点处存在附加的(离散对称，它与群~：：~(更准确地说，正如我们将在本文中看到的子群在~×中的“退化程度越强，相应的对称群越宽”)。*在个别情况下，找到了对称算符的显式形式，它在函数实现中被证明类似于分数导数(一般情况)或无穷小算符的单项式(在特定点)。一般结果(使用与我们不同的术语)已发表在[3]中。从引用的文献中可以看出，对称算子的意义与下列情况有关：1)它们定义了e(A)之间的等价关系；2)它们允许我们在e(A)的表示空间中区分不变子空间(在e(A)是可约的情况下)；3)它们在群G上函数的调和分析中起主要作用[3]。
The term" elementary representation" arises in connection with elementary harmonics, that is, exponents on circles and straight lines. An elementary representation of a semisimple complex group G is defined as a representation of the group induced by an arbitrary character 0~(k) of a maximal solvable subgroup K and is naturally realized in a class of functions G/K. In this connection unitary representations" of the ftmdamentalseries" of Gel'fand-Naimark are obtained for a subsystem of characters~(k). But in the general case it is natural to define an elementary representation e (~) in the class of infinitely differentiable functions on G/K. It is possible to consider e (~) as the analytic continuation of the" fundamental series" of Gel'fand-Naimark. However, instead of the nfumdamental series" it is also possible to use the series of all irreducible finitedimensional (real) representations of a group G.Until recently comparatively little was known about the structure of an arbitrary (nonunitary) representation e (o~). Special interest in this question arose in connection with the study of group rings of finite or rapidly decreasing functions on a group G [2-4, 6, 7, 10, 12]. In this connection it was discovered that in the class of elementary representations a twofold symmetry exists. On the one hand in this class, the Weyl group acts naturally whose orbits correspond to orbits in the class of characters~,~(k) and whose transformations reduce to equivalence relations on representations e (~)(for representations of the" fundamental series n the existence of this type of symmetry was established in the initial papers of IM Gel'land and MA Naimark [5]); however, at individual~ singular" points such relations degenerate. On the other hand from [2, 3, 6, 7, 10] there exists at singular points additional (discrete symmetry which is connected with the group~::~(more precisely as we shall see in this paper with subgroups in~× the stronger the" degree of degeneracy~ at the singular point the broader the corresponding symmetry group).* In individual cases~ an explicit form for symmetry operators was found which in a functional realization turned out to be analogous of fraction derivation (the general case) or monomials of infinitesimal operators (at special points). General results (in different terminology than ours) have been published in [3]. As can be seen from the references cited the significance of symmetry operators is connected with the following circumstances: 1) they define equivalence relations among e (a); 2) they permit us to distinguish invariant subspaces in a space of representations of e (a)(in cases when it is reducible); 3) they play a principle role [3] in harmonic analysis of functions on a group G.
DOI：
--
发表时间：
2006
期刊：
影响因子：
--
作者：
Takuya Hosokawa;Shuichi Ohno;Hidenori Fujiwara;藤原 英徳;Hidenori Fujiwara;藤原英徳;Hidenori Fujiwara;H.Fujiwara;Hidenori Fujiwara;Hidenori Fujiwara;Hidenori Fujiwara;Hidenori Fujiwara;藤原 英徳;Hidenori Fujiwara
通讯作者：
Hidenori Fujiwara