  An important question with a rich history is the extent to which the
symplectic category is larger than the Kaehler category. Many interesting
examples of non-Kaehler symplectic manifolds have been constructed. However,
sufficiently large symmetries can force a symplectic manifold to be Kaehler. In
this paper, we solve several outstanding problems by constructing the first
symplectic manifold with large non-trivial symmetries which does not admit an
invariant Kaehler structure. The proof that it is not Kaehler is based on the
Atiyah-Guillemin-Sternberg convexity theorem. Using the ideas of this paper, C.
Woodward shows that even the symplectic analogue of spherical varieties need
not be Kaehler.
