Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras

Abstract

Considering the general linear Lie superalgebra $\mathfrak{gl}(m|n)=\mathfrak{gl}(m|n{)}_{\overline{\stackrel{ˉ}{0}}}\oplus \mathfrak{gl}(m|n{)}_{\overline{\stackrel{ˉ}{1}}}$ over $\mathbb{C}$, we first formulate a super version of Vust theorem associated with a principal nilpotent element $e\in \mathfrak{gl}(m|n{)}_{\overline{\stackrel{ˉ}{0}}}$. As an application of this theorem, we then obtain a Schur-Sergeev duality for principal finite $W$-superalgebras which is partially a super version of Brundan-Kleshchev's higher level Schur-Weyl duality established in \cite{BKl}

37 Pages. Final version accepted for publication in Journal of Algebra 673 (2025) 138-187