Unitary representations of group extensions. I
Unitary representations of group extensions. I
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DOI：
10.1007/bf02392428
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发表时间：
1958-12
期刊：
影响因子：
3.7
通讯作者：
G. Mackey
中科院分区：
文献类型：
--
作者：
G. Mackey
设(~)是可分的局部紧群。继续在我们的文献[11]和[12]中采用的惯例，我们将术语“(~”的连续么正表示简写为“(~的表示”)。如果X是(~的表示在适当意义上是“全已知”的)的真闭子群，则可以提出以下两个问题：(A)3的哪些表示(它是~的不可约表示的限制？(B)给定3的这样一个表示(如何构造(它是限制的)的所有不可约表示？当3(是否单位子群问题(A)有一个平凡的答案(除维度问题外)，且问题(B)本质上与确定(~？)的所有不可约表示相同。然而，对于3(的其他选择，问题(A)和(B)可以提供确定(的所有不可约表示为两个更容易接近的分量的问题的有用的分解。本文的主要目的是讨论问题(A)和(B)及其在3(为正态的特殊情况下)的表示的确定中的应用。实际上，我们会发现更方便的是处理我们确定3的表示的细微变化(它们在[12]的第195页定义的意义上是准等价的。对于其中3(不仅是正规的，而且是可交换的，并且其中~是3(和~/3的半直积)的特殊情况，(~/3(在我们的论文(1)中已概要地进行了本论文的大部分内容)本文中的材料已在以下各部分中作了概要的描述)：(A)1954年10月在巴黎举行的两次讲座，由《亨利·波因卡尔》主持。(B)由普林斯顿大学物理系主办、尤金·希金斯基金资助的关于小组陈述的十次系列讲座。(C)课程(D)1955年美国数学学会夏季会议上的一篇论文(摘要61-6-726 t)。芝加哥大学数学系已经发布了芝加哥大学课程的油印讲稿，国家接收中心可能会出版一卷包含在Celloque Henri Poincar6上发表的演讲文本的卷。
Let (~ be a separable locally compact group. Continuing the convention adopted in our papers [11] and [12] we shall abbreviate the term" continuous unitary representation of (~" to" representation of (~". If X is a proper closed subgroup of (~ whose representations are in a suitable sense" all known" one may pose the following two questions.(a) Which representations of 3 (are the restrictions to it of irreducible representations of~?(b) Given such a representation of 3 (how can one construct all irreducible representations of (~ of which it is the restriction? When 3 (is the identity subgroup question (a) has a trivial answer (apart from questions of dimension) and question (b) is essentially the same as that of determining all irreducible representations of (~? However, for other choices of 3 (, questions (a) and (b) can furnish a useful breakdown of the problem of determining all irreducible representations of (~ into two more accessible components. It is the primary purpose of this paper to discuss questions (a) and (b) and their application to the determination of the representations of (~ in the special case in which 3 (is normal. Actually we shall find it more convenient to deal with the slight variation in which we identify representations of 3 (which are quasi equivalent in the sense defined on page 195 of [12]. For the special case in which 3 (is not only normal but commutative and in which~ is a semi direct product of 3 (and (~/3 (this program has been carried out in outline in our paper (1) In large part the material in this paper has been described in outline in each of the following:(a) Two lectures given in Paris in October 1954 under the auspices of the" Colloque Henri Poincar6".(b) A series of ten lectures on group representations given under the auspices of the Princeton University physics department and supported by the Eugene Higgins fund.(c) A course in group representations given during the 1955 summer quarter at the University of Chicago.(d) A paper presented by title at the 1955 summer meeting of the American Mathematical Society (Abstract 61-6-726 t). Mimeographed lecture notes of the University of Chicago course have been issued by the University of Chicago mathematics department and it is possible that the Centre Nationale des Recherehes Seientifiques will publish a volume containing the texts of the lectures presented at the" Celloque Henri Poincar6".