  In this note the interrelations between several natural morphisms on the
$\pi_1$ of groups of Hamiltonian diffeomorphisms are investigated. As an
application, the equality of the (non-linear) Maslov index of loops of
quantomorphisms of prequantizations of $\C P^n$ and the Calabi-Weinstein
invariant is shown, settling affirmatively a conjecture by A. Givental. We also
prove the proportionality of the mixed action-Maslov morphism and the Futaki
invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds, as
suggested by C. Woodward. Finally, a family of generalized action-Maslov
invariants is computed for toric manifolds via barycenters of their moment
polytopes, with an application to mass-linear functions recently introduced by
D. McDuff and S. Tolman.

Symplectic Geometry;Algebraic Topology;Differential Geometry