The Capelli identity, the double commutant theorem, and multiplicity-free actions
The Capelli identity, the double commutant theorem, and multiplicity-free actions
复制标题
卡佩利恒等式、双交换定理和无多重性作用
DOI：
10.1007/bf01459261
复制
发表时间：
1991
影响因子：
1.4
通讯作者：
T. Umeda
中科院分区：
文献类型：
--
作者：
R. Howe;T. Umeda
0. Capelli恒等式[Cal-3；W, p. 39]是经典不变理论中最著名和最有用的公式之一[W；D;CL;Z]。二重交换定理[W， p. 91]同样是结合代数一般理论中的一个基本结果。两者在Weyl的书中都扮演着关键的角色：古典团体。本文的主要目的是在向量空间上群的无多重行为EKJ的背景下证明两者之间的密切联系。我们讨论的重点是具有无多重作用的交换微分算子的结构。应用包括Shimura [Shl]和Rubenthaler和Schiffmann [RS]关于与埃尔米对称空间相关的b函数的一些公式的新推导，以及gin的通用包皮代数中心的有趣生成集的构造。我们还详细讨论了无多重表示的某些方面。这是论文内容的概述。在第一节中，我们将回顾经典的卡佩利恒等式。我们观察到它们的存在是如何通过双对易结果（1.6）和（1.8）来预测的，并且我们展示了如何使用它们来计算b函数。我们注意到Capelli明白他的算符产生由极化算符产生的代数的中心（见[B1，第77页]）。卡普利引入他的算子的动机是“解释”凯利的一个公式，本质上是b函数的计算。卡普利的观点是非常现代和结构主义的，在某些方面甚至比魏尔更现代。第2节至第9节提供了一个概念性背景，以理解卡佩利身份是一种无多样性行为的特征。它们包含了~ G的结构的一般讨论，多项式系数微分算子的代数，它们与给定的线性变换G群交换。定理9.1总结了讨论的结果，该定理特别指出多项式系数微分算子与a可交换
0. The Capelli identity [Cal-3; W, p. 39] is one of the most celebrated and useful formulas of classical invariant theory [W; D; CL; Z]. The double commutant theorem [W, p. 91] is likewise a basic result in the general theory of associative algebras. Both play key roles in Weyl's book: The classical groups. The main purpose of this paper is to demonstrate a close connection between the two, in the context of multiplicity-free actions EKJ of groups on vector spaces. The focus of our discussion will be the structure of the differential operators which commute with a multiplicity-free action. Applications include a new derivation of some formulas of Shimura [Shl ] and Rubenthaler and Schiffmann [RS] for b-functions associated to Hermitian symmetric spaces, and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin. We also give a detailed discussion of certain aspects of multiplicity-free representations. Here is an overview of the contents of the paper. In Sect. 1 we review the classical Capelli identities. We observe how their existence is predicted by a double commutant result (1.6) and (1.8), and we show how they can be used to compute b-functions. We remark that Capelli understood that his operators generate the center of the algebra generated by the polarization operators (see [B1, p. 77]). Also Capelli's motivation in introducing his operators was to "explain" a formula of Cayley essentially the computation of a b-function. Capelli's point of view was remarkably modern and structuralist, in certain ways more modern even than that of Weyl. Sections 2 through 9 provide a conceptual context for understanding the Capelli identities as a feature of multiplicity-free actions. They contain a general discussion of the structure of ~ G , the algebra of polynomial coefficient differential operators which commute with a given group G of linear transformations. The results of the discussion are summarized in Theorem 9.1, which says in particular that the polynomial coefficient differential operators commuting with a