Are two “nice” transformation groupoids with the same coarse moduli and isomorphic inertia isomorphic?

Hi!

I am stuck with the following question: suppose we have a semisimple connected algebraic group acting on a quasi-affine variety X by closed orbits, and suppose that inertia is flat. Suppose we have another quasi-affine variety Y which embeds inside X, and assume the normalizer in G of Y, say N, is a reductive algebraic group such that Y has closed orbits under its action. Assume that

$Y//N\cong X//G$

and that the restriction of the inertia stack from $[X/G]$ to $[Y//N]$ gives the inertia of the latter stack. Is it true that

$[Y/N]\cong [X/G]$

via the obvious morphism induced by the embedding $Y\hookrightarrow X$?

To fix ideas, one could consider

$X=\mathfrak{g}_{reg,ss}$

the regular semisimple points of the Lie algebra of $G$, the action being the adjoint, $Y=\mathfrak{t}_{reg}$ the regular points of a Cartan subalgebra of $\mathfrak{g}$ and $N=N(T)$ the normalizer of the corresponding torus. This example may be somehow misleading, as in this case inertia is abelian, so it descends uniquely to the coarse moduli giving a gerbe structure, but this may be false in the more general situation.

Thanks!

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