Finite $\mathcal{W}$-algebras of ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{1\mathrm{\mid }2n}$ and Ghost centers

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