The primitive spectrum for gl(m|n)

Kevin Coulembier, Ian M. Mussson

Published 2014-09-08Version 1

We study inclusions between primitive ideals for the general linear superalgebra g=gl(m|n). If k is a semisimple Lie algebra, then any primitive ideal in U(k) is the annihilator of a simple highest weight module, and the same is true for primitives in U(g). It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For k this quasi-order is essentially the left Kazhdan-Lusztig (KL) order and we derive an alternative definition of this order which extends to classical Lie superalgebras. We show that a relation in the KL order implies an inclusion between primitive ideals. For gl(m|n) the new KL quasi-order is defined explicitly in terms of Brundan's Kazhdan-Lusztig theory. We prove that the quasi-order induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of gl}(m|n) and in general for gl(2|2). An important tool is a new translation principle for primitive ideals, based on the crystal structure for category O. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for gl(m|1).