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.