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

Source abstract

Abstract We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over ${\mathbb {C}}$ 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 ${\mathcal {O}}$ . 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 $AIII$ .