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

Chih-Whi Chen ; Shun-Jen Cheng

Forum of Mathematics, Sigma, Tome 12 (2024)

Résumé

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 $AIII$.

Détail

DOI : 10.1017/fms.2024.17