The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

Abstract

AbstractWe solve two problems in representation theory for the periplectic Lie superalgebra$\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.