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

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