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Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre$\mathcal{A}$ of its enveloping algebra, which consists of certain elementsthat square to the center. We seek to generalize and enrich the anticentre tothe context of supersymmetric pairs $(\mathfrak{g},\mathfrak{k})$, or moregenerally supersymmetric spaces $G/K$. We define certain invariantdistributions on $G/K$, which we call ghost distributions, and which in somesense are induced from invariant distributions on $G_0/K_0$. Ghostdistributions, and in particular their Harish-Chandra polynomials, giveinformation about branching from $G$ to a symmetric subgroup $K'$ which isrelated (and sometimes conjugate) to $K$. We discuss the case of $G\times G/G$for an arbitrary quasireductive supergroup $G$, where our results prove theexistence of a polynomial which determines projectivity of irreducible$G$-modules. Finally, a generalization of Gorelik's ghost centre is definedcalled the full ghost centre, $\mathcal{Z}_{full}$. For type I basic Liesuperalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and provethat if $\mathfrak{g}$ contains an internal grading operator,$\mathcal{Z}_{full}$ consists exactly of those elements in$\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on everyfinite-dimensional irreducible representation.

Ghost distributions on supersymmetric spaces I: Koszul induced superspaces, branching, and the full ghost centre