Completeness of determinantal Hamiltonian flows on the matrix affine Poisson space

TL;DR

This paper proves the completeness of Hamiltonian flows for minors and Gelfand-Zeitlin systems on the matrix affine Poisson space, extending known results and enabling analytic Hamiltonian actions.

Contribution

It establishes the completeness of Hamiltonian flows for all minors and Gelfand-Zeitlin systems on the matrix affine Poisson space, generalizing previous classical results.

Findings

Hamiltonian flows of all minors are complete.

Gelfand-Zeitlin systems induce complete flows.

Extends classical flow completeness results to quadratic Poisson structures.

Abstract

The matrix affine Poisson space (M_{m,n}, pi_{m,n}) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m=n reduces to the standard Poisson structure on GL_n(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan-Zelevinsky integrable systems on M_{n,n} are complete and thus induce (analytic) Hamiltonian actions of C^{n(n-1)/2} on (M_{n,n}, pi_{n,n}) (as well as on GL_n(C) and on SL_n(C)). We define Gelfand-Zeitlin integrable systems on (M_{n,n}, pi_{n,n}) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure pi_{n,n} of the recent result of Kostant and Wallach [KW] that the flows of the complexified classical Gelfand-Zeitlin integrable systems are complete.