Quantum characteristic classes, moment correspondences and the Hamiltonian groups of coadjoint orbits

For any coadjoint orbit $G/L$, we determine all useful terms of the associated Savelyev-Seidel morphism defined on $H_{-*}(\Omega G)$. Immediate consequences are: (1) the dimension of the kernel of the natural map $\pi_*(G)\otimes \mathbb{Q}\rightarrow \pi_*(Ham(G/L))\otimes \mathbb{Q}$ is at most the semi-simple rank of $L$, and (2) the Bott-Samelson cycles in $\Omega G$ which correspond to Peterson elements are solutions to the min-max problem for Hofer's max-length functional on $\Omega Ham(G/L)$. The proof is based on Bae-Chow-Leung's recent computation of Ma'u-Wehrheim-Woodward morphism for the moment correspondence associated to $G/T$ where $T$ is a maximal torus, the computation of Abbondandolo-Schwarz isomorphism for $G$, and two theoretical results including the coincidence of the above Savelyev-Seidel and Ma'u-Wehrheim-Woodward morphisms, and a Leray-type spectral sequence relating Savelyev-Seidel morphisms for $G/L$ and $G/T$. These ingredients also allow us to obtain an alternative proof of Peterson-Woodward's comparison formula which relates the quantum cohomology of $G/T$ to that of $G/L$.