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Peterson conjecture via Lagrangian correspondences and wonderful compactifications

For a simply-connected compact semisimple Lie group G and its maximal torus T , we study the A∞-functor associated to the moment Lagrangian correspondence from the cotangent bundle T ∗G to the square G/T− × G/T . In particular, we compute the leading term of the A∞-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 ∗(∆,∆) of the diagonal ∆ in the square G/T− × 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 ∗(∆,∆) are isomorphic as rings after a localization.