实现洛伦兹群在德西特空间上的单元表征

IF 0.5 4区 数学 Q3 MATHEMATICS
Jan Frahm , Karl-Hermann Neeb , Gestur Ólafsson
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引用次数: 0

摘要

本文在前人研究的基础上,证明了对于作用于非紧因果对称空间M=G/H上的所有连通半单线性李群G, G的每一个不可约酉表示都可以通过作用于M上的向量束分布截面上的全纯扩展的边值映射来实现。本文讨论了作用于de Sitter空间M=dSd上的连通洛伦兹群G=SO1,d(R)e的这一过程。我们特别证明了先前构造的实子空间网络满足局部性条件。根据1990年代Bros和Moschella的思想,我们证明了矩阵值球函数对应于我们的可拓过程,解析可拓到复化群G =SO1,d()中的一个大域G切,对于d=1,它专指复切平面(−∞,0)。具体讨论了一些特殊情况:(A) d=1的情况,它与Hilbert空间中的标准子空间密切对应;(b)标量值函数的情况,对于d>;2是球表示的情况,我们也描述了de Sitter空间中切上全纯扩展的跳点;(c) d=3的情况,我们得到了矩阵值球函数的相当显式的公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Realization of unitary representations of the Lorentz group on de Sitter space
This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups G acting on a non-compactly causal symmetric space M=G/H, every irreducible unitary representation of G can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over M. In the present paper we discuss this procedure for the connected Lorentz group G=SO1,d(R)e acting on de Sitter space M=dSd. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain Gcut in the complexified group G=SO1,d(), which for d=1 specializes to the complex cut plane (,0]. A number of special situations is discussed specifically: (a) The case d=1, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for d>2 is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case d=3, where we obtain rather explicit formulas for the matrix-valued spherical functions.
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来源期刊
CiteScore
1.20
自引率
16.70%
发文量
74
审稿时长
79 days
期刊介绍: Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.
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