## Abstract

Let g=gl

_{N}(k), where k is an algebraically closed field of characteristic p>0, and N∈ℤ≥1. Let χ∈g^{∗}and denote by Uχ(g) the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional U_{χ}(g)-module has dimension divisible by p^{dχ}, where d_{χ}is half the dimension of the coadjoint orbit of χ. Our main theorem gives a classification of U_{χ}(g)-modules of dimension p^{dχ}. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for U_{0}(h) for a certain Levi subalgebra h of g; we view this as a modular analogue of Mœglin's theorem on completely primitive ideals in U(gl_{N}(ℂ)). To obtain these results, we reduce to the case χ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted W-algebra.Original language | English |
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Pages (from-to) | 1594-1617 |

Number of pages | 24 |

Journal | Compositio Mathematica |

Volume | 155 |

Issue number | 8 |

Early online date | 11 Jul 2019 |

DOIs | |

Publication status | Published - 1 Aug 2019 |

## Keywords

- general linear Lie algebras
- reduced enveloping algebras
- Finite W-algebras

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