## 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 |
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Article number | 108024 |

Journal | Advances in Mathematics |

Volume | 392 |

Early online date | 20 Sep 2021 |

DOIs | |

Publication status | Published - 3 Dec 2021 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)