The Seidel morphism of cartesian products

We prove that the Seidel morphism of $(M\times M^{\mathrm{\prime }},\omega \oplus {\omega }^{\mathrm{\prime }})$ is naturally related to the Seidel morphisms of $(M,\omega )$ and $(M^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})$, when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with non-trivial image via Seidel’s morphism, leads to an injection of the fundamental group of the group of Hamiltonian diffeomorphisms of the other component into the fundamental group of the group of Hamiltonian diffeomorphisms of the product. This result was inspired by and extends results obtained by Pedroza [P08].