Let $X$ be a smooth affine variety with an algebraic symplectic form $\omega$. Let $G$ be a finite subgroup of the group of symplectomorphisms of $X$.

We can say that $X$ is trivially a normal variety with symplectic singularities, so we are in the hypothesis of Theorem $2.3$ in Kaledin's paper https://arxiv.org/pdf/math/0310186.pdf.

I would like to confirm or refute the following:

Are the connected components of the isotropy types $\{X_H\}_{H\leq G}$, where $H$ is a stabiliser of some point $x\in X$, the symplectic leaves of the Poisson scheme $(X, \omega^{-1})$? In particular, do the connected components of the isotropty types coincide with the Poisson strata in Theorem $2.3$ of https://arxiv.org/pdf/math/0310186.pdf?



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