2021/01/27 by Alexander Sherman, Sherman, Alexander
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2101.11285
openalex publication_date 2021/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Lie superalgebra \mathfrakg, 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 (\mathfrakg,\mathfrakk), 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 called the full ghost centre, Zfull. For type I basic Lie superalgebras \mathfrakg we fully describe Zfull, and prove that if \mathfrakg contains an internal grading operator, Zfull consists exactly of those elements in U\mathfrakg acting by ℤ-graded constants on every finite-dimensional irreducible representation.