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

Abstract

Given a Lie superalgebra $$\mathfrak {g}$$ g , Gorelik defined the anticentre $$\mathcal {A}$$ 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 $$(\mathfrak {g},\mathfrak {k})$$ ( 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 $$G_0/K_0$$ 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 ′ which is related (and sometimes conjugate) to K. We discuss the case of $$G\times G/G$$ 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, $$\mathcal {Z}_{full}$$ Z full . For type I basic Lie superalgebras $$\mathfrak {g}$$ g we fully describe $$\mathcal {Z}_{full}$$ Z full , and prove that if $$\mathfrak {g}$$ g contains an internal grading operator, $$\mathcal {Z}_{full}$$ Z full consists exactly of those elements in $$\mathcal {U}\mathfrak {g}$$ U g acting by $$\mathbb {Z}$$ Z -graded constants on every finite-dimensional irreducible representation.