  We quantize the problem considered by Bott-Samelson who applied Morse theory
to compact symmetric spaces $G/K$ and the associated real flag manifolds
$\Lambda$. The starting point is the construction of a Lagrangian
correspondence generalizing Weinstein's moment correspondence, from the
cotangent bundle of $G/K$ to the complexification of $\Lambda$. Then
Ma'u-Wehrheim-Woodward's quilted Floer theory gives an $A_{\infty}$
homomorphism $\Phi:C_{-*}(\Omega(G/K))\rightarrow CF^*(\Lambda,\Lambda)$ from
the dga of singular chains on the based loop space $\Omega(G/K)$ to the Floer
cochain complex of $\Lambda$. Our main result is reminiscent of the famous
conjecture of Peterson about the quantum cohomology of complex flag varieties:
$H^*(\Phi)$ becomes an isomorphism after localization. It is a generalization
of a result proved by Bae-Leung on the case when $G/K$ is a Lie group.
