The Modality of a Borel Subgroup in a Simple Algebraic Group of Type E8

Let G be a simple algebraic group over an algebraically closed field k, where char k is either 0 or a good prime for G. We consider the modality mod (B:u) of the action of a Borel subgroup B of G on the Lie algebra u of the unipotent radical of B, and report on computer calculations used to show that mod (B:u)=20, when G is of type E8. This completes the determination of the values for mod (B:u) for G of exceptional type.