Some closed formulas for super Kazhdan-Lusztig polynomials

Kazhdan--Lusztig theory for finite-dimensional representations of the general linear Lie superalgebra $\mathfrak{gl}(m|n)$ is well understood through the Khovanov arc-diagram combinatorics. On the other hand, the Kazhdan--Lusztig conjecture for the full super category $\mathcal O$ was established by Cheng--Lam--Wang and Brundan--Losev--Webster, and the corresponding Kazhdan--Lusztig polynomials can be computed by Brundan's algorithm. The aim of this paper is to deepen our understanding of super category $\mathcal O$ and its Kazhdan--Lusztig polynomials by bringing together three ingredients: auxiliary modules associated with changes of Borel subalgebras introduced by Cheng--Lam--Wang, the Koszul grading studied by Brundan--Losev--Webster, and Whittaker coinvariants functor investigated by Brundan--Goodwin. We obtain, in particular, closed formulas for certain Kazhdan--Lusztig polynomials associated with odd reflections, determine the socles of some auxiliary modules, and realize certain pullbacks of standard modules via parabolic Miura transforms. We also use Khovanov arc-diagram combinatorics to characterize certain regular dominant weights intrinsically in terms of the underlying abelian category. We discuss how this viewpoint can be used to obtain concrete examples relevant to the study of Morita equivalence classes of blocks of super category $\mathcal O$.