Given a Lie superalgebra g g , Gorelik defined the anticentre A A of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs (g,k) ( g , k ) , or more generally supersymmetric spaces G / K . We define certain invariant distributions on G / K , which we call ghost distributions, and which in some sense are induced from invariant distributions on G₀/K₀ G 0 / K 0 . Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup $$K'$$ K ′ which is related (and sometimes conjugate) to K . We discuss the case of G× G/G G × G / G for an arbitrary quasireductive supergroup G , where our results prove the existence of a polynomial which determines projectivity of irreducible G -modules. Finally, a generalization of Gorelik’s ghost centre is defined which we call the full ghost centre, Zfull Z full . For type I basic Lie superalgebras g g we fully describe Zfull Z full , and prove that if g g contains an internal grading operator, Zfull Z full consists exactly of those elements in Ug U g acting by Z Z -graded constants on every finite-dimensional irreducible representation.
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Alexander Sherman (2024) studied this question.