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.
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Acknowledgments
I would like to thank the anonymous referee for making many helpful suggestions to improve the clarity of this article, and for explaining the simple argument for Theorem 8.2. I would also like to thank Vanessa Miemietz and Shaun Stevens at the University of East Anglia for many useful discussions, and Max Nazarov for hosting me at the University of York. The research leading to these results has received funding from the European Commission, Seventh Framework Programme, under Grant Agreement No. 600376, as well as Grant CPDA125818/12 from the University of Padova.
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Topley, L.W. A Morita theorem for modular finite W-algebras. Math. Z. 285, 685–705 (2017). https://doi.org/10.1007/s00209-016-1724-8
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DOI: https://doi.org/10.1007/s00209-016-1724-8