A Morita theorem for modular finite W-algebras

TL;DR

This paper establishes a Morita equivalence between finite W-algebras and central reductions of universal enveloping algebras for Lie algebras over fields of large positive characteristic, extending Skryabin's equivalence to the modular setting.

Contribution

It proves a modular Morita theorem linking finite W-algebras to central reductions of universal enveloping algebras, generalizing Skryabin's equivalence to positive characteristic.

Findings

Finite W-algebras are Morita equivalent to central reductions of universal enveloping algebras.

The result extends Skryabin's equivalence to modular Lie algebra settings.

Provides a new perspective on the structure of modular finite W-algebras.

Abstract

We consider the Lie algebra $\mathfrak{g}$ of a simple, simply connected algebraic group over a field of large positive characteristic. For each nilpotent orbit $\mathcal{O} \subseteq \mathfrak{g}$ we choose a representative $e\in \mathcal{O}$ and attach a certain filtered, associative algebra $\widehat{U}(\mathfrak{g},e)$ known as a finite $W$-algebra, defined to be the opposite endomorphism ring of the generalised Gelfand-Graev module associated to $(\mathfrak{g}, e)$. This is shown to be Morita equivalent to a certain central reduction of the enveloping algebra of $U(\mathfrak{g})$. The result may be seen as a modular version of Skryabin's equivalence.