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We prove that the finite $\mathcal{W}$-algebra associated to$\mathfrak{osp}_{1|2n}$ and its principal nilpotent element is isomorphic toGorelik's ghost center of $\mathfrak{osp}_{1|2n}$, which proves an analog ofKostant's theorem for $\mathfrak{osp}_{1|2n}$.

Finite $\mathcal{W}$-algebras of $\mathfrak{osp}_{1|2n}$ and Ghost centers