  We construct the moduli space of finite dimensional representations of
generalized quivers for arbitrary connected complex reductive groups using
Geometric Invariant Theory as well as Symplectic reduction methods. We explicit
characterize stability and instability for generalized quivers in terms of
Jordan-H\"older and Harder Narasimhan objects, reproducing well-known results
for classical case of quiver representations. We define and study the Hesselink
and Morse stratifications on the parameter space for representations, and
bootstrap them to an inductive formula for the equivariant Poincar\'e
Polynomial of the moduli spaces of representations. We work out explicitly the
case of supermixed quivers, showing that it can be characterized in terms of
slope conditions, and that it produces stability conditions different from the
ones in the literature. Finally, we resolve the induction of Poincar\'e
polinomials for a particular family of orthogonal representations.
