  Consider a Hamiltonian action of a compact Lie group H on a compact
symplectic manifold (M,w) and let G be a subgroup of the diffeomorphism group
Diff(M). We develop techniques to decide when the maps on rational homotopy and
rational homology induced by the classifying map BH --> BG are injective. For
example, we extend Reznikov's result for complex projective space CP^n to show
that both in this case and the case of generalized flag manifolds the natural
map H_*(BSU(n+1)) --> H_*(BG) is injective, where G denotes the group of all
diffeomorphisms that act trivially on cohomology. We also show that if lambda
is a Hamiltonian circle action that contracts in G = Ham(M,w) then there is an
associated nonzero element in pi_3(G) that deloops to a nonzero element of
H_4(BG). This result (as well as many others) extends to c-symplectic manifolds
(M,a), ie, 2n-manifolds with a class a in H^2(M) such that a^n is nonzero. The
proofs are based on calculations of certain characteristic classes and
elementary homotopy theory.
