Simple character formulas for finite $W$-superalgebras of type $A$

We establish a canonical basis character formula for the irreducible modules in arbitrary parabolic BGG-type categories, including the category of finite-dimensional modules, for finite $W$-superalgebras of type $A$. These categories categorify the tensor product modules of irreducible polynomial representations and their duals over a quantum group of type $A$. Moreover, the standard modules and irreducible modules in these categories categorify the standard basis and Lusztig’s dual canonical basis in the tensor product modules. Our formula provides a uniform generalization of several character formulas in BGG categories for Lie superalgebras and for $W$-algebras of type $A$.