Translation for finite 𝑊-algebras

Goodwin, Simon

doi:10.1090/S1088-4165-2011-00388-5

Translation for finite $W$-algebras
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by Simon M. Goodwin

Represent. Theory 15 (2011), 307-346

DOI: https://doi.org/10.1090/S1088-4165-2011-00388-5

Published electronically: April 5, 2011

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Abstract:

A finite $W$-algebra $U(\mathfrak {g},e)$ 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 $\mathfrak {g}$. It is possible to give the tensor product of a $U(\mathfrak {g},e)$-module with a finite dimensional $U(\mathfrak {g})$-module the structure of a $U(\mathfrak {g},e)$-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 $U(\mathfrak {g},e)$.

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