We study two types of actions on King's moduli spaces of quiver representations over a

field $k$, and we decompose their fixed loci using group cohomology in order to give

modular interpretations of the components. The first type of action arises by considering

finite groups of quiver automorphisms. The second is the absolute Galois group of a

perfect field $k$ acting on the points of this quiver moduli space valued in an algebraic

closure of $k$; the fixed locus is the set of $k$-rational points, which we decompose

using the Brauer group of $k$, and we describe the rational points as quiver representations

over central division algebras over $k$. Over the field of complex numbers, we describe the

symplectic and holomorphic geometry of these fixed loci in hyperk\"ahler quiver varieties

using the language of branes. This is joint work with Florent Schaffhauser.

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