Harmonic Analysis on Symmetric Spaces―Higher Rank Spaces, Positive Definite Matrix Space and Generalizations
Harmonic Analysis on Symmetric Spaces―Higher Rank Spaces, Positive Definite Matrix Space and Generalizations
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对称空间的调和分析——高阶空间、正定矩阵空间及其推广
DOI：
10.1007/978-1-4939-3408-9
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发表时间：
2016
期刊：
影响因子：
3.8
通讯作者：
A. Terras
中科院分区：
文献类型：
--
作者：
A. Terras
本文介绍了对称空间的调和分析，重点关注高阶空间、正定矩阵空间和泛化等高级主题。它适用于数学专业的研究生或物理或工程研究人员。与标题为“对称空间的调和分析 - 欧几里得空间、球面和庞加莱上半平面”的介绍性书籍一样，该书的风格是非正式的，强调动机、具体示例、历史和应用。这里考虑的对称空间是商 X= G/K，其中 G 是非紧实李群，例如所有 nxn 非奇异实数的一般线性群 GL (n, P)矩阵，以及 K= O (n)，正交矩阵的最大紧子群和四元数上半“平面”。在一般线性群的情况下，可以将 X 识别为 nxn 正定对称矩阵的空间 Pn。新版本中纳入了许多修正和更新，包括对随机矩阵理论和量子混沌的讨论，以及关于模形式及其相应的最新研究。添加了许多更高阶的 L 函数，例如 Pn 上的热方程的解、Donald St. P. Richards 对 Pn 的中心极限定理、欧几里得空间中球体的最密集晶格堆积的结果，以及平面域中拉普拉斯特征值的 Weyl 定律的 GL (n) 类似物。离散群 Г 的 X（例如具有整数项和行列式±1 的 nxn 矩阵的模群 GL (n, Z)），与在欧几里得空间中寻找球体最稠密格堆积的问题、自同构形式、Hecke 算子、L 函数和 Selberg 迹公式及其在谱论和数论中的应用的联系。
This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled" Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X= G/K, where G is a non-compact real Lie group, such as the general linear group GL (n, P) of all nxn non-singular real matrices, and K= O (n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half" plane" and the quaternionic upper half" plane". In the case of the general linear group, one can identify X with the space Pn of nxn positive definite symmetric matrices. Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank. Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL (n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups Γ (such as the modular group GL (n, Z) of nxn matrices with integer entries and determinant±1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.