For a simply-connected compact semisimple Lie group $G$ and its maximal torus $T$, we study the $A_{ \infty}$-functor associated to the moment Lagrangian correspondence from the cotangent bundle $T^*G$ to the square $G/T^{ -} \times G/T$. In particular, we compute the leading term of the $A_{ \infty}$-homomorphism from the wrapped Floer cohomology $HW^*(T^*_e G, T^*_e G)$ of the cotangent fiber $T_e^*G$ to the Floer cohomology $HF^*(\Delta, \Delta)$ of the diagonal $\Delta$ in the square $G/T^{ -} \times G/T$ by determining the count of certain pseudo-holomorphic quilts. As a consequence, we prove that the Floer cohomologies $HW^*(T^*_e G, T^*_e G)$ and $HF^*(\Delta,\Delta)$ are isomorphic as rings after a localization.
Categories: math.SG