We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group G. We first study the case when G = O(V ) or Sp(V ), interpreting them as orthogonal (resp. symplectic) bundle representations of the symmetric quivers introduced by Derksen-Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. Finally, we discuss Hitchin-Kobayashi correspondences for these objects.this can be summarized as follows.Proposition 1.1. Let Q be a quiver. A twisting for Q is equivalent to a principal GL(α)-bundle F C together with vector spaces M α . A twisted representation of Q with dimension vector is equivalent to the prescription of a principal GL(i)-bundle E C together with a finite-dimensional representation space Rep(Q, V ), and a section φ ∈ Ω 0 ((E C × F C ) × ρ Rep(Q, V )).From our notation, it is already obvious that we are interested in passing to maximal compacts, by endowing our vector bundles with hermitian metrics. Indeed, a hermitian metric on V i corresponds to a reduction h i ∈ A 0 (E C i (GL i /U i )) of the structure group of E C i to the unitary group U i := U(V i ).

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-Hölder 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 Poincaré 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 Poincaré polinomials for a particular family of orthogonal representations.

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