A note on Humphreys' conjecture on blocks

TL;DR

This paper confirms Humphreys' conjecture on blocks of reduced enveloping algebras under weaker assumptions and introduces a new proof approach for type G2 in characteristic 3, supported by dimensional calculations.

Contribution

It extends the validity of Humphreys' conjecture to weaker conditions and provides a novel proof method for type G2 in characteristic 3.

Findings

Humphreys' conjecture holds under weaker assumptions.

A new proof approach for type G2 in characteristic 3.

Dimensional calculations for centralisers in exceptional Lie algebras.

Abstract

Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping algebras $U_\chi({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a reductive algebraic group over an algebraically closed field of characteristic $p>0$ and $\chi\in{\mathfrak g}^{*}$. It is well-known to hold under Jantzen's standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result for ${\mathfrak g}$ of type $G_2$ in characteristic 3, a case in which the previously mentioned weaker assumptions do not hold. This approach requires some dimensional calculations for certain centralisers, which we conduct in the Appendix for all the exceptional Lie algebras in bad characteristic.