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

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 $W(e)\to U({\mathfrak{gl}}_n)$, 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 [M].

We have revised the statement and proof of Theorem 3.10

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