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

TL;DR

This paper establishes a connection between Verma modules and Whittaker modules for quasi-reductive Lie superalgebras, providing a complete classification of their composition factors and linking to quantum symmetric pairs.

Contribution

It introduces a functorial approach that relates Verma modules to Whittaker modules and categorifies structures associated with quantum symmetric pairs in Lie superalgebras.

Findings

Backelin functor maps Verma modules to Whittaker modules when they exist

Complete classification of composition factors of standard Whittaker modules

Categorification of q-symmetrizing maps and Fock spaces for quantum symmetric pairs

Abstract

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 $\mathcal O$. 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 $AIII$.