# The moduli space of generalized quivers

## TL;DR

This paper constructs and analyzes the moduli space of generalized quiver representations for complex reductive groups, introducing new stability conditions and stratifications, and providing explicit formulas for their topological invariants.

## Contribution

It develops a comprehensive framework for the moduli space of generalized quivers, including stability criteria, stratifications, and explicit Poincaré polynomial formulas, extending classical results.

## Key findings

- Explicit characterization of stability and instability conditions.
- Development of Hesselink and Morse stratifications for generalized quivers.
- Derivation of inductive formulas for equivariant Poincaré polynomials.

## Abstract

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.