The action of a compact Lie group, G, on a compact sympletic manifold
gives rise to some remarkable combinatorial invariants. The simplest and
most interesting of these is the moment polytope, a convex polyhedron
which sits inside the dual of the Lie algebra of G. One of the main
goals of this monograph is to describe what kinds of geometric
information are encoded in this polytope. For instance, the first
chapter is largely devoted to the Delzant theorem, which says that there
is a one-one correspondence between certain types of moment polytopes
and certain types of symplectic G-spaces. (One of the most challenging
unsolved problems in symplectic geometry is to determine to what extent
Delzant's theorem is true of every compact symplectic G-Space.)
The moment polytope also encodes quantum information about the actions
of G. Using the methods of geometric quantization, one can frequently
convert this action into a representations, p, of G on a Hilbert
space, and in some sense the moment polytope is a diagrammatic picture
of the irreducible representations of G which occur as
subrepresentations of p. Precise versions of this item of folklore are
discussed in Chapters 3 and 4. Also, midway through Chapter 2 a more
complicated object is discussed: the Duistermaat-Heckman measure, and
the author explains in Chapter 4 how one can read off from this measure
the approximate multiplicities with which the irreducible
representations of G occur in p. This gives an excuse to touch on
some results which are in themselves of great current interest: the
Duistermaat-Heckman theorem, the localization theorems in equivariant
cohomology of Atiyah-Bott and Berline-Vergne and the recent extremely
exciting generalizations of these results by Witten, Jeffrey-Kirwan,
Lalkman, and others.
The last two chapters of this book are a self-contained and somewhat
unorthodox treatment of the theory of toric varieties in which the usual
hierarchal relation of complex to symplectic is reversed. This book is
addressed to researchers
