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

Considering the general linear Lie superalgebra $\mathfrak{gl}(m|n)=\mathfrak{gl}(m|n)_{\bar{\bar 0}}\oplus \mathfrak{gl}(m|n)_{\bar{\bar 1}}$ over $\mathbb{C}$, we first formulate a super version of Vust theorem associated with a principal nilpotent element $e\in \mathfrak{gl}(m|n)_{\bar{\bar 0}}$. 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}