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## Abstract

Let G be connected reductive algebraic group defined over an algebraically closed field of characteristic p>0 and suppose that p is a good prime for the root system of G, the derived subgroup of G is simply connected and the Lie algebra g=Lie(G) admits a non-degenerate (AdG)-invariant symmetric bilinear form. Given a linear function χ on g we denote by U
_{χ}(g) the reduced enveloping algebra of g associated with χ. By the Kac–Weisfeiler conjecture (now a theorem), any irreducible U
_{χ}(g)-module has dimension divisible by p
^{d(χ)} where 2d(χ) is the dimension of the coadjoint G-orbit containing χ. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra U
_{χ}(g) admits a module of dimension p
^{d(χ)}.

Original language | English |
---|---|

Article number | 108024 |

Journal | Advances in Mathematics |

Volume | 392 |

Early online date | 20 Sept 2021 |

DOIs | |

Publication status | Published - 3 Dec 2021 |

### Bibliographical note

Funding Information:Acknowledgement. Part of this work was done in Spring 2018 when the first author was in residence at MSRI (Berkeley). He would like to thank the Institute for the hospitality and support. The work of the second author is funded by the UKRI Future Leaders fellowship program, grant number MR/S032657/1 .

## Keywords

- Humphreys' conjecture
- Reduced enveloping algebras
- Reductive groups
- Representations of Lie algebras
- Small modules

## ASJC Scopus subject areas

- Mathematics(all)

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