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Let (G, V) be a representation with either G a torus or (G, V) a locally freestable $\theta$-representation. We study the fiber at 0 of the associatedmoment map, which is a commuting variety in the latter case. We characterizethe cases where this fiber is normal. The quotient (i.e. the symplecticreduction) turns out to be a specific orbifold when the representation ispolar. In the torus case, this confirms a conjecture stated by C. Lehn, M.Lehn, R. Terpereau and the author in a former article. In the $\theta$-case,the conjecture was already known but the present approach yield another proof.

On the normality of the null-fiber of the moment map for $\theta$- and tori representations