A Morita theorem for 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.