  We introduce a class of $G$-invariant connections on a homogeneous principal
bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space
carries the structure of normal variety and has a canonical anti-holomorphic
involution. The fixed points of the anti-holomorphic involution are precisely
the integrable invariant complex structures on $Q.$ This normal variety is
closely related to quiver varieties and, more generally, to varieties of
commuting matrix tuples modulo simultaneous conjugation.
