Ghost distributions on supersymmetric spaces II: basic classical superalgebras

TL;DR

This paper explores ghost distributions on supersymmetric spaces associated with basic classical Lie superalgebras, introducing interlaced pairs and a ghost algebra, and analyzing their properties and representations.

Contribution

It introduces the concept of interlaced pairs, defines a ghost algebra for these pairs, and studies their realization as equivariant operators and the properties of the Harish-Chandra morphism.

Findings

Defined interlaced pairs with Iwasawa decompositions.

Constructed a ghost algebra generalizing Gorelik's subalgebra.

Proved injectivity of the Harish-Chandra morphism and computed its image for rank one cases.

Abstract

We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $(\mathfrak{g},\mathfrak{k})$ and $(\mathfrak{g},\mathfrak{k}')$ admit Iwasawa decompositions. For such pairs we define a ghost algebra, generalizing the subalgebra of $\mathcal{U}\mathfrak{g}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the `special' ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $(\mathfrak{g},\mathfrak{k})$ is interlaced.