Lax formalism for Gelfand-Tsetlin integrable systems

Abstract

In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems defined on (co)adjoint orbits of the compact Lie groups $\mathrm{U}(n)$ and $\mathrm{SO}(n)$. In the particular setting of (co)adjoint orbits of $\mathrm{U}(n)$, by means of the associated Lax matrix we construct a family of algebraic curves which encodes the Gelfand-Tsetlin integrable systems as branch points. This family of algebraic curves enables us to explore some new insights into the relationship between the topology of singular Gelfand-Tsetlin fibers, singular algebraic curves and vanishing cycles. Further, we provide a new description for Guillemin and Sternberg's action coordinates in terms of hyperelliptic integrals.

A second main result (Theorem 2) and a new section (Section 5) were added. Minor modifications in sections 3 and 4; 31 pages, 14 figures