  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 $G/L$.
