  Extending our reduction construction in \cite{Hu} to the Hamiltonian action
of a Poisson Lie group, we show that generalized K\"ahler reduction exists even
when only one generalized complex structure in the pair is preserved by the
group action. We show that the constructions in string theory of the
(geometrical) $T$-duality with $H$-fluxes for principle bundles naturally arise
as reductions of factorizable Poisson Lie group actions. In particular, the
group may be non-abelian.
