On the algebraic independence of Hamiltonian characteristic classes (Q719902)

Let \((M, \omega)\) be a closed symplectic manifold of dimension \(2n\) and \(\text{Ham} (M, \omega)\) the group of Hamiltonian diffeomorphisms of \((M, \omega)\). Consider the characteristic classes of a Hamiltonian fibration \(\mu_k(E) := \pi !(\Omega^{n+k}) \in H^{2k}(B)\). The fundamental question is whether these classes are nontrivial or not and to what extent they are algebraically independent in the cohomology ring \(H^\star(B \operatorname{Ham}(M, \omega))\) of the classifying space of the group of Hamiltonian diffeomorphisms. A. G. Reznikov proved the algebraic independence for \(\mathbb CP^n\) and also suggested that this result may be true for any adjoint orbit of a compact Lie group. In this paper, the authors prove that Reznikov's claim is true generically (see Theorem 1.1). In Section 3, the authors also provide some examples of coadjoint orbits exhibiting cases where algebraic independence does hold or not. In Section 4, they present an application to lattices in semisimple groups.