The primitive spectrum and category O for the periplectic Lie superalgebra

TL;DR

This paper advances the understanding of the representation theory of the periplectic Lie superalgebra pe(n) by describing its primitive spectrum and decomposing its category O into blocks using novel functorial methods.

Contribution

It introduces a new equivalence between category O for classical Lie superalgebras and Harish-Chandra bimodules, and establishes a BGG reciprocity for pe(n).

Findings

Primitive spectrum characterized via braid group functors.

Category O decomposed into indecomposable blocks.

New equivalence between category O and Harish-Chandra bimodules.

Abstract

We solve two problems in representation theory for the periplectic Lie superalgebra pe(n), namely the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category O into indecomposable blocks. To solve the first problem we establish a new type of equivalence between category 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.