Typical representations of Takiff superalgebras

Abstract

We investigate representations of the $\ell$-th Takiff superalgebras $\widetilde{\mathfrak g}_\ell := \widetilde{\mathfrak g}\otimes \mathbb C[\theta]/(\theta^{\ell+1})$, for $\ell>0$, associated with a basic classical and a periplectic Lie superalgebras $\widetilde{\mathfrak g}$. We introduce the odd reflections and formulate a general notion of typical representations of the Takiff superalgebras $\widetilde{\mathfrak g}_\ell$. As a consequence, we provide a complete description of the characters of the finite-dimensional modules over type I Takiff superalgebras. For the Lie superalgebras $\widetilde{\mathfrak g}= \mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$, we prove that the Kac induction functor of $\widetilde{\mathfrak g}_\ell$ leads to an equivalence from an arbitrary typical Jordan block of the category $\mathcal O$ for $\widetilde{\mathfrak g}_\ell$ to a Jordan block of the category $\mathcal O$ for the even subalgebra of $\widetilde{\mathfrak g}_\ell$. We also obtain a classification of non-singular simple Whittaker modules over the Takiff superalgebras.