Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras

Abstract

For a basic classical Lie superalgebra $\mathfrak{s}$, let $\mathfrak{g}$ be the central extension of the Takiff superalgebra $\mathfrak{s}\otimes \Lambda (\theta )$, where $\theta$ is an odd indeterminate. We study the category of $\mathfrak{g}$-Whittaker modules associated with a nilcharacter $\chi$ of $\mathfrak{g}$ and show that it is equivalent to the category of $\mathfrak{s}$-Whittaker modules associated with a nilcharacter of $\mathfrak{s}$ determined by $\chi$. In the case when $\chi$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak{s}$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra.

27 pages