  We prove that the linearization functor from the category of Hamiltonian
K-actions with group-valued moment maps in the sense of Lu, to the category of
ordinary Hamiltonian K-actions, preserves products up to symplectic
isomorphism. As an application, we give a new proof of the Thompson conjecture
on singular values of matrix products and extend this result to the case of
real matrices. We give a formula for the Liouville volume of these spaces and
obtain from it a hyperbolic version of the Duflo isomorphism.
