The Duflo-Serganova functor, vingt ans après

Abstract

We review old and new results concerning the DS functor and associated varieties for Lie superalgebras. These notions were introduced in the unpublished manuscript [DS] by Michel Duflo and the third author. This paper includes the results and proofs of the original manuscript, as well as a survey of more recent results.

On the Duflo-Serganova functor for the queer Lie superalgebra

“…The other check that needs to be made is that for stable weights λ ∈ P + (q n ), µ ∈ P + (q n ), and for t ∈ C, we have However following the argument of Lemma 6.5.4, we see that the main step is to take an eigenspace of a certain semisimple operator z, which clearly commutes with h since they both lie in t. Thus Lemma 6.5.4 becomes, in our case, [DS x (L(λ)) : L gx (ν) t ] = [DS x (L g ′ ×h ′′ (λ)) : L g ′ x ×h ′′ (ν) t ]. Now using the statement of Lemma 6.5.5 along with this, we obtain Equation (15).…”

“…The DS-functor was introduced in [7]. We recall definitions and some results of [7], [15] in the Appendix. A study of the DS-functor for q n when x 2 = 0 was initiated in [27], see also [11], Section 5 for the proofs.…”

“…The DS-functor was introduced in [7]; see also [15] for an expanded exposition. We will be using a slight extension of DS-functor which we define below.…”

“…For a finite-dimensional g-module we denote by [N] its image in K − (g). Although DS x is not an exact functor, by the Hinich lemma DS x defines ring homomorphisms on reduced Grothendieck rings K − (F in(g)) → K − (F in(g x )) which coincides with the restriction map [M] → [Res g g x M], see [15]. The ring K − (F in(g x )) is a subring of K − (F in(g x )) and the image of K − (F in(g)) → K − (F in(g x )) lies in this subring, see [15].…”

“…is often denoted by ds x in many papers, including [15]. We chose to avoid this notation to emphasize the simplicity and universality of restriction, in favor of the more limited setting of the DS functor.…”