On the Duflo-Serganova functor for the queer Lie superalgebra

Abstract

We study the Duflo-Serganova functor DS x for the queer Lie superalgebra q n and for all odd x with [x, x] semisimple. For the case when the rank of x is 1 we give a formula for multiplicities in terms of the arch diagram attached to λ. Further, we prove that DS x (L) is semisimple if L is a simple finite-dimensional module and x is of rank 1 satisfying x 2 = 0.

Jacobson-Morozov Lemma for algebraic supergroups

“…Proof. The proof is essentially identical to the computation of DS x given in [GS22], and we will explain it using the language from that article. Namely, because DS k x,y is a symmetric monoidal functor, and it takes the standard module to the standard module, it will commute with translation functors.…”

“…For the q(n) case we use the computations of [GS22]. Given any simple g x = g ymodule L ′ of integral weight, it is shown there that for any two composition factors of DS x L or DS y L isomorphic to L ′ , the difference of their h-weights must be even.…”

“…Arc diagrams. We recall from [GS22] the arc diagrams associated to half-integral weights. First of all, the simple half-integral weight modules for q(n) are indexed by their highest weights, which are given by a strictly decreasing sequence of half-integers, i.e.…”

“…We say that a position • in an arc diagram is free if it is not the end of any arc. Given a dominant half-integral weight λ and a half-integer n/2, define ℓ(λ, n/2) to be the number of free positions to the left of n/2 in the arc diagram of λ (see [GS22] for examples).…”

“…Further, it commutes with the operation of removing core symbols for stable weights, because this is given by taking the eigenspace of a semisimple element z which commutes with both x and y. Thus following the algorithm described in [GS22], we obtain the formula…”