Finite W-superalgebras for basic Lie superalgebras

Abstract: We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic Lie superalgebra ${\ggg}\bbf=(\ggg\bbf)\bz+(\ggg\bbf)\bo$ associated with a nilpotent element $e\in (\ggg\bbf){\bar0}$ both over the field of complex numbers $\bbf=\mathbb{C}$ and over $\bbf={\bbk}$ an algebraically closed field of positive characteristic. In this paper, we mainly present the PBW theorem for $U({\ggg}\bbf,e)$. Then the construction of $U({\ggg}\bbf,e)$ can be understood well, which in contrast with finite $W$-algebras, is divided into two cases in virtue of the parity of $\text{dim}\,\mathfrak{g\bbf}(-1)_{\bar1}$. This observation will be a basis of our sequent work on the dimensional lower bounds in the super Kac-Weisfeiler property of modular representations of basic Lie superalgebras (cf. \cite[\S7-\S9]{ZS}).