Ghost Distributions on Supersymmetric Spaces I: Koszul Induced Superspaces, Branching, and the Full Ghost Centre

Given a Lie superalgebra g, Gorelik defined the anticentre 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), 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. Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup K^′ which is related (and sometimes conjugate) to K. We discuss the case of 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, Z_full. For type I basic Lie superalgebras g we fully describe Z_full, and prove that if g contains an internal grading operator, Z_full consists exactly of those elements in Ug acting by Z-graded constants on every finite-dimensional irreducible representation.