Finite $W$-superalgebras and the dimensional lower bounds for the representations of basic Lie superalgebras

Abstract

In this paper we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\mathfrak{g}_\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type $A$. Under the assumption of this conjecture, we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable. Such lower bounds, as a super-version of Kac-Weisfeiler conjecture, were formulated by Wang-Zhao in \cite{WZ} for the modular representations of a basic Lie superalgebra ${\ggg}_{\bbk}$ over an algebraically closed field $\bbk$ of positive characteristic $p$.