Integrable system on minimal nilpotent orbit

TL;DR

This paper constructs a complete integrable system on minimal nilpotent orbits for all complex simple Lie algebras using Schubert divisors, connecting to Coulomb branch systems and Gelfand-Zeitlin functions.

Contribution

It introduces a new integrable system on minimal nilpotent orbits applicable to all complex simple Lie algebras, including exceptional types, with explicit Hamiltonian computations.

Findings

Revealed integrable systems on minimal nilpotent orbits for classical and exceptional Lie algebras.

Connected the system to Gelfand-Zeitlin functions for classical types.

Computed Hamiltonian functions for each Dynkin diagram vertex in exceptional types.

Abstract

We show that for every complex simple Lie algebra, the equations of Schubert divisors on the flag variety give a complete integrable system of the minimal nilpotent orbit. The approach is motivated by the integrable system on Coulomb branch. 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.