Let $\mathfrak{ g}=\mathfrak{ g}_{ \bar0}+\mathfrak{ g}_{ \bar1}$ be a basic classical Lie superalgebra over $\mathbb{ C}$, and $e=e_{ \theta}\in\mathfrak{ g}_{ \bar0}$ with $-\theta$ being a minimal root of $\mathfrak{ g}$. Set $U(\mathfrak{ g},e)$ to be the minimal finite $W$-superalgebras associated with the pair $(\mathfrak{ g},e)$. In this paper we study the highest weight theory for $U(\mathfrak{ g},e)$, introduce the Verma modules and give a complete isomorphism classification of finite-dimensional irreducible modules, via the parameter set consisting of pairs of weights and levels. Those Verma modules can be further described via parabolic induction from Whittaker modules for $\mathfrak{ osp}(1|2)$ or $\mathfrak{ sl}(2)$ respectively, depending on the detecting parity of $\textsf{ r}:=\dim\mathfrak{ g}(-1)_{ \bar1}$. We then introduce and investigate the BGG category $\mathcal{ O}$ for $U(\mathfrak{ g},e)$, establishing highest weight theory, as a counterpart of the works for finite $W$-algebras by Brundan-Goodwin-Kleshchev and Losev, respectively. In comparison with the non-super case, the significant difference here lies in the situation when $\textsf{ r}$ is odd, which is a completely new phenomenon. The difficulty and complicated computation arise from there.
Categories: math.RT