The Primitive Spectrum and Category  for the Periplectic Lie Superalgebra

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.2010 Mathematics Subject Classification. 16E30, 17B10.

Tilting modules for classical Lie superalgebras

“…For a simple reflection , we have the right exact twisting functor on as in [11, §4.3], see also [1, 2]. Since these functors satisfy the braid relations, see, for example, [20, 30], we have the twisting functor defined via composition with respect to an arbitrary reduced expression for .…”

“…Moreover, its inverse (as described in the proof of [21, Theorem 8.1]) is by [1, Theorem 4.1], a cohomology functor of a completion functor. In [11, Section 4.2], it is shown that completion functors can be defined on as well and satisfy the analogue of (). We denote the corresponding cohomology functor (both for and ) by .…”

“…Moreover, its inverse (as described in the proof of[21, Theorem 8.1]) is by [1, Theorem 4.1], a cohomology functor of a completion functor. In[11, Section 4.2]…”

Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras

“…For simple Lie superalgebras, the analogue of Duflo's theorem is established by Musson in [Mu1] using the results of finite ring extensions in [Le1]. Following methods in [Le1,Mu1], it is shown in [CC,Section 4.1] that Duflo's theorem remains valid for arbitrary quasireductive Lie superalgebras. Therefore, it is natural to make an identification between annihilator ideals of simple Whittaker modules and primitive ideals from the category O.…”

“…These assumptions and choice will be referred to as (A1), (A2) and (A3) in this article; see also examples in (2.3.2). Lie superalgebras g satisfying these assumptions are referred to as Lie superalgebras of type I-0 (see [CC]) and fit into the framework of [CC,Section 4]. In particular, for such Lie superalgebras there is a number of basic properties of the twisting functors developed in [CC,Section 4.3] that are to be used in the present paper.…”