Super duality for Whittaker modules and finite $W$-algebras

TL;DR

This paper establishes a super duality equivalence between categories of Whittaker modules and finite W-algebras for classical Lie algebras and superalgebras, extending to the infinite-rank limit.

Contribution

It introduces a novel super duality framework linking Whittaker modules and W-algebras for Lie superalgebras, generalizing previous results to the infinite-rank setting.

Findings

Super duality as an equivalence of module categories

Extension of super duality to finite W-algebras and superalgebras

Application of Losev-Shu-Xiao decomposition in the proof

Abstract

We establish a super duality as an equivalence between Whittaker module categories over a pair of classical Lie algebra and Lie superalgebra in the infinite-rank limit. Building on this result and utilizing the Losev-Shu-Xiao decomposition, we obtain a super duality which is an equivalence between module categories over a pair of finite $W$-algebras and $W$-superalgebras at the infinite-rank limit.