On the normality of the null-fiber of the moment map for θ- and tori representations

Let (G, V) be a representation with either G a torus or (G, V) a locally free stable $\theta$-representation. We study the fiber at 0 of the associated moment map, which is a commuting variety in the latter case. We characterize the cases where this fiber is normal. The quotient (i.e. the symplectic reduction) turns out to be a specific orbifold when the representation is polar. 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.