Modular representations of Lie algebras of reductive groups and Humphreys' conjecture

Alexander Premet, Lewis Topley

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Article number108024
JournalAdvances in Mathematics
Early online date20 Sep 2021
Publication statusPublished - 3 Dec 2021


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

ASJC Scopus subject areas

  • Mathematics(all)


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