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12 mins ago comment added Balazs This is likely to be subtle. Craw-Maclagan-Thomas in arxiv.org/abs/math/0611840 constructed an example of a finite subgroup $G\subset GL(3)$ and a monomial equivariant ideal $J$ such that $[J] \in G$-Hilb does not lie on the "main" coherent component, so in particular $G$-Hilb is not irreducible. I am not sure about connectedness even in this example.
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