  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.
