Quantizations of nilpotent orbits vs 1-dimensional representations of W-algebras

TL;DR

This paper explores the relationship between quantizations of nilpotent orbits in semisimple Lie algebras and 1-dimensional modules of W-algebras, establishing a one-to-one correspondence that clarifies their connection.

Contribution

It demonstrates a one-to-one correspondence between quantizations of nilpotent orbits and 1-dimensional W-modules, clarifying a relationship previously identified by Moeglin.

Findings

Establishes a bijection between orbit quantizations and 1-dimensional W-modules

Clarifies the relationship originally discovered by Moeglin in the 1980s

Provides a framework for understanding quantizations of nilpotent orbits

Abstract

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0 and O be a nilpotent orbit in g. Then Orb is a symplectic algebraic variety and one can ask whether it is possible to quantize $\Orb$ (in an appropriate sense) and, if so, how to classify the quantizations. On the other hand, for the pair (g,O) one can construct an associative algebra W called a (finite) W-algebra. The goal of this paper is to clarify a relationship between quantizations of O (and of its coverings) and 1-dimensional W-modules. In the first approximation, our result is that there is a one-to-one correspondence between the two. The result is not new: it was discovered (in a different form) by Moeglin in the 80's.