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Harmonic Analysis on Symmetric Spaces-Higher Rank Spaces, Positive Definite Matrix Space and Generalizations (Paperback, Softcover reprint of the original 2nd ed. 2016)
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Harmonic Analysis on Symmetric Spaces-Higher Rank Spaces, Positive Definite Matrix Space and Generalizations (Paperback, Softcover reprint of the original 2nd ed. 2016)
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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 Poincare 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 n x n 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 n x n 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 (such as the
modular group GL(n,Z) of n x n 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.
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