infinite dimensional analogs of Gelfand pairs. That elegant area is extremely active
but its level of technicality takes it out of the scope of this book.
Acknowledgment s
Much of the material in Parts 1, 2 and 3 was the subject of courses I taught at
the University of California, Berkeley, over a period of years. Questions, comments
and suggestions from participants in those courses certainly improved the exposi-
tion. Some of the material in Part 3 relies on earlier treatments of J. Dieudonne [Di]
and J. Faraut [Fa], and much of the material in Part 4 depends on O. Yakimova's
doctoral dissertation [Y3]. In addition, a number of mathematicians looked at early
versions of this book and made useful suggestions. These include D. Akhiezer (com-
munications concerning his work with E. B. Vinberg on weakly symmetric spaces),
D. Bao (discussions on Finsler manifolds), R. Goodman (advice on how to organize
a book), I. A. Latypov and V. M. Gichev (communications concerning their work
on invariant function algebras), J. Lauret, H. Nguyen and G. Olafsson (for going
over the manuscript), G. Ratcliff and C. Benson (communications concerning their
work with J. Jenkins on spherical functions for commutative Heisenberg nilmani-
folds), and the three mathematicians who refereed this volume (for some very useful
I especially want to thank O. Yakimova for a number of helpful conversations
concerning her work and E. B. Vinberg's work on classification of smooth commu-
tative spaces.
Notational Conventions
M, C, M and O denote the real, complex, quaternionic and octonionic number
systems. If F is one of them, then x H- X* denotes the conjugation of F over R,
m x n
denotes the space o f m x n matrices over F, and if x G F
m x n
then x* e F
n X m
is its conjugate transpose. We write R e F
n x n
for the hermitian (x = x*) elements of
F n x n and ReFp Xn for those of trace 0, and we write ImF n X n for the skew-hermitian
(x + #* = 0) elements of F n X n ; that corresponds to the case n = 1.
In general we use upper case roman letters for groups, and when possible we
use the corresponding lower case letters for their elements. If G is a Lie group then
g denotes its Lie algebra. If I) is a Lie subalgebra of g then (unless it is defined
differently) H is the corresponding analytic subgroup of G.
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