Typical representations of Takiff superalgebras

We investigate representations of the $\mathrm{\ell }$-th Takiff superalgebras ${\tilde{\mathfrak{g}}}_{\mathrm{\ell }}:=\tilde{\mathfrak{g}}\otimes \mathbb{C}[\theta ]\mathrm{/}({\theta }^{\mathrm{\ell }+1})$, for $\mathrm{\ell }>0$, associated with a basic classical and a periplectic Lie superalgebras $\tilde{\mathfrak{g}}$. We introduce the odd reflections and formulate a general notion of typical representations of the Takiff superalgebras ${\tilde{\mathfrak{g}}}_{\mathrm{\ell }}$. 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 $\tilde{\mathfrak{g}}=\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$ and $\mathfrak{o}\mathfrak{s}\mathfrak{p}(2\mathrm{\mid }2n)$, we prove that the Kac induction functor of ${\tilde{\mathfrak{g}}}_{\mathrm{\ell }}$ leads to an equivalence from an arbitrary typical Jordan block of the category $\mathcal{O}$ for ${\tilde{\mathfrak{g}}}_{\mathrm{\ell }}$ to a Jordan block of the category $\mathcal{O}$ for the even subalgebra of ${\tilde{\mathfrak{g}}}_{\mathrm{\ell }}$. We also obtain a classification of non-singular simple Whittaker modules over the Takiff superalgebras.