The primitive spectrum of basic classical Lie superalgebras

TL;DR

This paper proves a conjecture describing primitive ideal inclusions for general linear superalgebras and introduces Kazhdan-Lusztig orders for basic classical Lie superalgebras, linking them to categorical braid group actions.

Contribution

It confirms the conjecture for general linear superalgebras and develops new Kazhdan-Lusztig quasi-orders for basic classical Lie superalgebras, relating them to primitive spectrum descriptions.

Findings

Primitive spectrum described by Kazhdan-Lusztig orders

Conjecture 5.7 proved for general linear superalgebra

Two Kazhdan-Lusztig orders are shown to be identical under certain conditions

Abstract

We prove Conjecture 5.7 in [arXiv:1409.2532], describing all inclusions between primitive ideals for the general linear superalgebra in terms of the Ext1-quiver of simple highest weight modules. For arbitrary basic classical Lie superalgebras, we formulate two types of Kazhdan-Lusztig quasi-orders on the dual of the Cartan subalgebra, where one corresponds to the above conjecture. Both orders can be seen as generalisations of the left Kazhdan-Lusztig order on Hecke algebras and are related to categorical braid group actions. We prove that the primitive spectrum is always described by one of the orders, obtaining for the first time a description of the inclusions. We also prove that the two orders are identical if category O admits `enough' abstract Kazhdan-Lusztig theories. In particular, they are identical for the general linear superalgebra, concluding the proof of the conjecture.