Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras
Changjie Cheng, Bin Shu, Yang Zeng
TL;DR
The paper establishes a super version of Vust's theorem for $\mathfrak{gl}(m|n)$ with a principal nilpotent $e$, and derives a Schur-Sergeev duality for principal finite $W$-superalgebras by connecting centralizer algebras to a super degenerate cyclotomic Hecke algebra acting on $V^{\otimes d}$. It develops a framework around Dynkin and Kazhdan filtrations, pyramid descriptions of $\mathfrak{g}_e$, and Skryabin's equivalence to relate $W_\chi$-modules with Whittaker $\mathfrak{g}$-modules, enabling a contraction picture where $U(\mathfrak{g}_e)$ arises as a degeneration of $W'_\chi$. The main result is a higher level Schur-Sergeev duality in the super setting: End$_{H_d(\Lambda)}(V^{\otimes d})$ equals the image of $W_\chi$, and End$_{W_\chi}(V^{\otimes d})^{op}$ equals $H_d(\Lambda)^{op}$, generalizing Brundan-Kleshchev's duality via a graded/contraction approach. This work provides a robust super-geometry and representation-theory framework for double centralizers, with potential categorial and combinatorial interpretations for principal and refined $W$-superalgebras.
Abstract
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}