The Primitive Spectrum of a Basic Classical Lie Superalgebra

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 Ext 1 -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.

The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

“…For Lie superalgebras of type I, we prove that whenever O admits a suitable duality, twisting and completion functors are isomorphic up to conjugation with this duality, as is well-known in various specific cases, see [KM, CM]. In particular, this allows us to express the primitive spectrum for pe(n) in terms of twisting functors, as an extension of the main result of [Co1].…”

“…Proof. We adapt the proof of [Co1,Theorem 5.3]. In view of Corollaries 3.2 and 3.3 and Proposition 3.5, it suffices to show that L(M0 λ , L 2 ) is a subquotient of…”

“…The description of the primitive spectrum of a reductive Lie algebra is a classical result, with the last piece of the proof obtained in [Vo], see [Mu2,Section 15.3] for an overview. In [Co1] it was proved that the primitive spectrum of a simple basic classical Lie superalgebra can be described in terms of the combinatorics of the twisting functors on category O. For the case gl(m|n) this even led to a description of the primitive spectrum in terms of the Ext 1 -quiver of category O.…”

“…For the case gl(m|n) this even led to a description of the primitive spectrum in terms of the Ext 1 -quiver of category O. An essential ingredient in the construction in [Co1] was the equivalence between O and a category of Harish-Chandra bimodules, see [BG, MM].…”

Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras

“…For E ∈ F , we recall that E ⊗ U is equipped with a natural g-bimodule structure as in [BG,Section 2.2] and [Co,Section 2.4]:…”