Whittaker modules and representations of finite $W$-algebras of queer Lie 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 $\mathfrak{q}(n)$.

28 pages