Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra

TL;DR

This paper classifies irreducible cuspidal modules over isplaystylesl_{n+1} by relating them to finite-dimensional modules over a minimal nilpotent finite W-algebra, providing explicit realizations without complex localization methods.

Contribution

It demonstrates that all finite-dimensional irreducible W(e)-modules are quotients of isplaystylegl_n-modules and constructs explicit realizations of cuspidal modules avoiding previous complex techniques.

Findings

All irreducible W(e)-modules are quotients of isplaystylegl_n-modules.

Explicit realizations of cuspidal isplaystylesl_{n+1}-modules are provided.

Avoids using twisted localization and coherent families for construction.

Abstract

A weight $\mathfrak{sl}_{n+1}$-module with finite-dimensional weight spaces is called a cuspidal module, if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively on it. In \cite{LL}, it has been shown that any block with a generalized central character of the cuspidal $\mathfrak{sl}_{n+1}$-module category is equivalent to a block of the category of finite-dimensional modules over the minimal nilpotent finite $W$-algebra $W(e)$ for $\mathfrak{sl}_{n+1}$. In this paper, using a centralizer realization of $W(e)$ and an explicit embedding $W(e)\rightarrow U(\mathfrak{gl}_n)$, we show that every finite-dimensional irreducible $W(e)$-module is isomorphic to an irreducible $W(e)$-quotient module of some finite-dimensional irreducible $\mathfrak{gl}_n$-module. As an application, we can give very explicit realizations of all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules using finite-dimensional irreducible $\mathfrak{gl}_n$-modules, avoiding using the twisted localization method and the coherent family introduced in \cite{M}.