A note on Humphreys' conjecture on blocks

Humphreys' conjecture on blocks parametrises the blocks of a reduced enveloping algebra of the Lie algebras of a reductive algebraic group over an algebraically closed field of positive characteristic. 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 in 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.