Duflo-Serganova fumctors and Brundan-Goodwin's parabolic inductions

Duflo--Serganova functors play an important role in the representation theory of Lie superalgebras. While it is desirable to understand the images of modules under DS, little is known beyond finite-dimensional representations. For general linear Lie superalgebras, Brundan--Goodwin study the Whittaker coinvariants functor $H_{0}$ and the associated principal $W$-superalgebra. In this paper we investigate rank-one DS functors attached to odd roots, characterized by the condition that $\operatorname{DS}_{x}(\mathfrak g)\subset \mathfrak g$ is a graded subsuperalgebra with respect to the principal good grading, and the induced functors $\overline{\operatorname{DS}}$ on $W$-superalgebra module categories via the Skryabin equivalence. In particular, we explicitly compute the DS images of $\mathfrak b$-Verma supermodules (for a suitable class of Borel subalgebras $\mathfrak b$). We also observe that, via the parabolic Miura transform, the pullbacks of tensor products of (dual) Verma modules for the $W$-superalgebra can be identified with the $H_{0}$-images of $\mathfrak b$-Verma supermodules for an appropriate choice of $\mathfrak b$.