Semisimplicity of the $DS$ functor for the orthosymplectic Lie superalgebra

Abstract

We prove that the Duflo-Serganova functor DS x attached to an odd nilpotent element x of osp(m|2n) is semisimple, i.e. sends a semisimple representation M of osp(m|2n) to a semisimple representation of osp(m − 2k|2n − 2k) where k is the rank of x. We prove a closed formula for DS x (L(λ)) in terms of the arc diagram attached to λ.

The Duflo-Serganova functor, vingt ans après

“…Serganova originally conjectured that these functors are semisimple when g is basic classical, meaning that they takes semisimple modules to semisimple modules. Following the work of [HsW] and [GH1] this is now a theorem. For p(n) these functors are known not to be semisimple, while for q(n) this remains an open question.…”

“…More recently, Ehrig and Stroppel have done similar work on realizing Rep OSp(m|2n) as a certain diagram algebra, (see [EhSt1] and [EhSt2]). Their diagram algebra is related to type D Khovanov algebras; however, their arc diagrams differ from those used in [GH1] to study the action of DS x on simple modules. A dictionary to go between them is described in appendix A of [GH1].…”

“…Their diagram algebra is related to type D Khovanov algebras; however, their arc diagrams differ from those used in [GH1] to study the action of DS x on simple modules. A dictionary to go between them is described in appendix A of [GH1].…”

“…osp(m|2n) case. For a full explanation of the osp case with many examples, see [GH1]. Below we closely follow the treatment given in [G4].…”

“…There are a number of parallels between the principal blocks of osp(2k+1|2k) and osp(2k+2|2k). In particular in [GH1] they find an explicit bijection τ between simple modules such that it respects the action of the DS functor, meaning we have an equality of multiplicity numbers [DS x…”