Endomorphismensatz for Lie superalgebras

Question

For semisimple complex Lie algebras there is Soergel's Endomorphismensatz

$$C = \operatorname{End}(P(w_0)) \cong \mathbf C[\mathfrak h]/\mathbf C[\mathfrak h]^W$$

for $w_0$ the longest element in the Weyl group $W$, and the Struktursatz which says that the functor

$$\begin{aligned} \mathbb V: \mathcal O_0 &\longrightarrow \operatorname{Mod-End}(P(w_0)),\\ M &\longmapsto \operatorname{Hom}(P(w_0), M) \end{aligned}$$

is fully faithful on projectives.

Question: Is there a similar statement for Lie superalgebras? I know that this paper, thm. 4.5 give a super-generalization for the Struktursatz. However, is there also an equivalent of the Endomorphismensatz?

Answer 1

Certainly, there's nothing as clean as Soergel's statement. The issue, as alluded to in the paper of Brundan you link is that there isn't just one anti-dominant highest weight in a given block of category O; there can be infinitely many. Thus, C must be replaced by a much bigger sum of projectives for the Endomorphismensatz to hold. In fact, if you look at an atypical block for $\mathfrak{gl}(1|1)$, every highest weight is anti-dominant, so the Endomorphismensatz only holds if you take with a projective that has every indecomposable projective as a summand!