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

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