Translation for finite W-algebras

TL;DR

This paper studies the properties of translation functors for finite W-algebras, which are important for understanding their representation theory, by examining how modules behave under tensoring with finite-dimensional modules.

Contribution

It establishes fundamental properties of translation functors for finite W-algebras, advancing the understanding of their module categories.

Findings

Translation functors preserve certain module structures.

Tensor products with finite-dimensional modules induce new modules.

Properties of translations are foundational for W-algebra representation theory.

Abstract

A finite $W$-algebra $U(\g,e)$ is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element $e$ of a complex reductive Lie algebra $\g$. It is possible to give the tensor product of a $U(\g,e)$-module with a finite dimensional $U(\g)$-module the structure of a $U(\g,e)$-module; we refer to such tensor products as translations. In this paper, we present a number of fundamental properties of these translations, which are expected to be of importance in understanding the representation theory of $U(\g,e)$.