Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs

We show that, for an arbitrary quasi-reductive Lie superalgebra with a triangular decomposition and a character zeta of the nilpotent radical, the associated Backelin functor Gamma(zeta) sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category O. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor Gamma(zeta) and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type AIII.