  From a view point of the moment map, we shall introduce the notion of
Einstein-Hermitian generalized connections over a generalized K\"ahler manifold
of symplectic type.
  We show that moduli spaces of Einstein-Hermitian generalized connections
arise as the K\"ahler quotients. The deformation complex of Einstein-Hermitian
generalized connections is an elliptic complex and it turns out that the smooth
part of the moduli space is a finite dimensional K\"ahler manifold. The
canonical line bundle over a generalized K\"ahler manifold of symplectic type
has the canonical generalized connection and its curvature coincides with "the
scalar curvature as the moment map" which is defined in the previous paper
[Goto_2016]. K\"ahler-Ricci solitons provide examples of Einstein-Hermitian
generalized connections and Einstein Hermitian co-Higgs bundles are also
discussed.
