Finite $W$-Superalgebras and Dimensional Lower Bounds for the Representations of Basic Lie Superalgebras

Abstract

In this paper we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable, under an assumption on the minimal dimensions of representations of the finite $W$-superalgebra $U({\mathfrak{g}}_{\mathbb{C}},e)$ over the field of complex numbers. The afore-mentioned lower bounds for modular representations, as a super-version of the Kac-Weisfeiler conjecture [26], were formulated and proved by Wang-Zhao in [35] for basic Lie superalgebras over an algebraically closed field $\mathbb{k}$ of positive characteristic $p$. We further conjecture that the assumption is actually satisfied (see Conjecture 1.2). This is to say, the complex finite $W$-superalgebra $U({\mathfrak{g}}_{\mathbb{C}},e)$ affords either one-dimensional or two-dimensional representations, according to the parity of the discriminant number (the difference of dimensions between the odd part of ${\mathfrak{g}}_{\mathbb{C}}$ and its subspace centralized by $e$). We demonstrate the positivity of the conjecture with examples including all the cases of type $A$, and finally reduce the investigation of the conjecture to the case of rigid nilpotent elements as the situation happens for the ordinary finite $W$-algebras (cf. [29]).