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

TL;DR

This paper introduces ghost distributions on supersymmetric spaces, generalizes Gorelik's anticentre to a full ghost centre, and explores their implications for representation theory and module projectivity in Lie superalgebras.

Contribution

It defines ghost distributions on supersymmetric spaces, generalizes the ghost centre concept, and describes the full ghost centre for type I basic Lie superalgebras, linking it to module actions.

Findings

Existence of a polynomial determining projectivity of irreducible modules.

Full description of the full ghost centre for type I basic Lie superalgebras.

Identification of the full ghost centre with elements acting by graded constants on irreducible representations.

Abstract

Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre $\mathcal{A}$ 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 $(\mathfrak{g},\mathfrak{k})$, 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 $G_0/K_0$. 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 $G\times G/G$ 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 called the full ghost centre, $\mathcal{Z}_{full}$. For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that 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 every finite-dimensional irreducible representation.