Equivariant Casson invariants via gauge theory

Let Σ be an integral homology 3-sphere with a finite cyclic group action making it into a branched cover of another integral homology sphere with branch set a knot K. We define a family of equivariant Casson invariants of Σ by essentially counting equivariant SU(2)-representations of its fundamental group. We relate these invariants to the Euler characteristic of 3-orbifold Floer homologies, and identify them with certain equivariant knot signatures of K. We give explicit formulae for the equivariant Casson invariants of the branched covers over the graph knots and Montesinos knots. Applying this to various natural cyclic group actions on the links of singularities, we obtain a geometric proof of the Fintushel-Stern and Neumann-Wahl formulae and their extensions, and give a closed form formula for the Floer homology of Brieskorn homology spheres.