Peterson conjecture via Lagrangian correspondences and wonderful compactifications

Abstract

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 $T^{\ast }G$ 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 $H{W}^{\ast }(T_e^{\ast }G,T_e^{\ast }G)$ of the cotangent fiber $T_e^{\ast }G$ to the Floer cohomology $H{F}^{\ast }(\Delta ,\Delta )$ 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 $H{W}^{\ast }(T_e^{\ast }G,T_e^{\ast }G)$ and $H{F}^{\ast }(\Delta ,\Delta )$ are isomorphic as rings after a localization.

Combined with arXiv:2103.00382v2