Integrable System on Minimal Nilpotent Orbit

We show that for every complex simple Lie algebra \(\mathfrak {g}\), the equations of Schubert divisors on the flag variety \(G/B^-\) give a complete integrable system of the minimal nilpotent orbit \(\mathcal {O}_{\min }\). The approach is motivated by the integrable system on Coulomb branch as reported by Braverman (arXiv preprint arXiv:1604.03625, 2016).We give explicit computations of these Hamiltonian functions, using Chevalley basis and a so-called Heisenberg algebra basis. For classical Lie algebras we rediscover the lower order terms of the celebrated Gelfand-Zeitlin system. For exceptional types we computed the number of Hamiltonian functions associated to each vertex of Dynkin diagram. They should be regarded as analogs of Gelfand-Zeitlin functions on exceptional type Lie algebras.