Some Remarks on CMV Matrices and Dressing Orbits

The CMV matrices are the unitary analogs of Jacobi matrices. In the finite case, it is well-known that the set of Jacobi matrices with a fixed trace is nothing but a coadjoint orbit of the lower triangular group. In this note, we will give the analog of this result for the CMV matrices. En route, we also discuss the Hamiltonian formulation of the Lax equations for the defocusing Ablowitz-Ladik hierarchy.