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The Primitive Spectrum of a Basic Classical Lie Superalgebra

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Abstract

We prove Conjecture 5.7 in Coulembier and Musson (Math. J., arXiv:1409.2532), describing all inclusions between primitive ideals for the general linear superalgebra in terms of the \({{\rm Ext}^{1}}\)-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 \({\mathcal{O}}\) admits ‘enough’ abstract Kazhdan–Lusztig theories. In particular, they are identical for the general linear superalgebra, concluding the proof of the conjecture.

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  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Kevin Coulembier

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Correspondence to Kevin Coulembier.

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Communicated by Y. Kawahigashi

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Coulembier, K. The Primitive Spectrum of a Basic Classical Lie Superalgebra. Commun. Math. Phys. 348, 579–602 (2016). https://doi.org/10.1007/s00220-016-2667-y

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