Whittaker modules for classical Lie superalgebras

TL;DR

This paper classifies simple Whittaker modules for classical Lie superalgebras, establishing equivalences with Harish-Chandra bimodules and reducing composition factor problems to well-understood Verma modules, enabling explicit calculations.

Contribution

It introduces a classification of simple Whittaker modules, establishes a categorical equivalence, and reduces composition factor analysis to Kazhdan-Lusztig combinatorics for certain superalgebras.

Findings

Classification of simple Whittaker modules for classical Lie superalgebras.

Establishment of Miličić-Soergel type equivalence with Harish-Chandra bimodules.

Explicit computation of composition series for modules over lgebra (m|n) and osp(2|2n) using Kazhdan-Lusztig theory.

Abstract

We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Mili\v{c}i\'c-Soergel equivalence of a category of Whittaker modules and a category of Harish-Chandra bimodules. For classical Lie superalgebras of type I, we reduce the problem of composition factors of standard Whittaker modules to that of Verma modules in their BGG categories $\mathcal O$. As a consequence, the composition series of standard Whittaker modules over the general linear Lie superalgebras $\mathfrak{gl}(m|n)$ and the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2|2n)$ can be computed via the Kazhdan-Lusztig combinatorics.