====== 2025-01-23 ======
  * Discussion with Xin Jin about her proof


===== Xin Jin's result =====
Let $G$ be a (adjoint or simply connected?) group. We consider $J^G_\gfrak$, the regular universal centralizer. What's the definition? 
  * I guess one definition is, take $T^* G$, consider the 

One definition is that double Whittaker reduction,