Highest Weight Theory for Minimal Finite $W$-Superalgebras and Related Whittaker Categories

Abstract

Let $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}+{\mathfrak{g}}_{\bar{1}}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_{\theta }\in {\mathfrak{g}}_{\bar{0}}$ 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 $\mathsf{r}:=\dim \mathfrak{g}(-1{)}_{\bar{1}}$. We then introduce and investigate the BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highest weight theory, as a counterpart for the works for finite $W$-algebras by Brundan–Goodwin–Kleshchev [Int. Math. Res. Not. IMRN 15 (2008), article no. rnn051] and Losev [in: Geometric methods in representation theory II (2012), 353–370]. In comparison with the non-super case, the significant difference here lies in the situation when $\mathsf{r}$ is odd, which is a completely new phenomenon. The difficulty and complicated computation arise from there.