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

Given a Lie superalgebra \(\mathfrak {g}\) $\mathfrak{g}$ , Gorelik defined the anticentre \(\mathcal {A}\) $\mathcal{A}$ 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 \((\mathfrak {g},\mathfrak {k})\) $(\mathfrak{g},\mathfrak{k})$ , 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 \(G_0/K_0\) $G_0/K_0$ . Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup \(K'\) $K^{\prime }$ which is related (and sometimes conjugate) to K. We discuss the case of \(G\times G/G\) $G\times G/G$ 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, \(\mathcal {Z}_{full}\) ${\mathcal{Z}}_{\mathit{full}}$ . For type I basic Lie superalgebras \(\mathfrak {g}\) $\mathfrak{g}$ we fully describe \(\mathcal {Z}_{full}\) ${\mathcal{Z}}_{\mathit{full}}$ , and prove that if \(\mathfrak {g}\) $\mathfrak{g}$ contains an internal grading operator, \(\mathcal {Z}_{full}\) ${\mathcal{Z}}_{\mathit{full}}$ consists exactly of those elements in \(\mathcal {U}\mathfrak {g}\) $\mathcal{U}\mathfrak{g}$ acting by \(\mathbb {Z}\) $\mathbb{Z}$ -graded constants on every finite-dimensional irreducible representation.