Whittaker modules and representations of finite $W$-algebras of queer Lie superalgebras

TL;DR

This paper explores Whittaker modules over queer Lie superalgebras, establishing category equivalences and reducing complex module problems to well-understood BGG category issues, thus advancing the representation theory of these superalgebras.

Contribution

It formulates standard Whittaker modules for queer Lie superalgebras and connects their structure to BGG categories, providing new tools for analyzing module composition and multiplicities.

Findings

Reduced composition factor analysis to BGG Verma modules

Established equivalences between Whittaker modules and category O

Extended Losev-Shu-Xiao decomposition to superalgebras

Abstract

We study various categories of Whittaker modules over the queer Lie superalgebras $\mathfrak q(n)$. We formulate standard Whittaker modules and reduce the problem of composition factors of these standard Whittaker modules to that of Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$. We also obtain an analogue of Losev-Shu-Xiao decomposition for the finite $W$-superalgebras $U(\mathfrak q(n), E)$ of $\mathfrak q(n)$ associated to an odd nilpotent element $E\in \mathfrak q(n)_{\bar{1}}$. As an application, we establish several equivalences of categories of Whittaker $\mathfrak q(n)$-modules and analogues of BGG category of $U(\mathfrak q(n), E)$-modules. In particular, we reduce the multiplicity problem of Verma modules over $U(\mathfrak q(n), E)$ to that of the Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$.