Ghost distributions on supersymmetric spaces II: basic classical superalgebras

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.