Left Cells in the Weyl Group of Type E8

The aim of the present paper is to give an explicit description for all the left cells of the Weyl group W of type E8. We first find a representative set Σ for the left cells of W and then use it to describe these cells. Our results show that most of the left cells in W can be characterized by their generalized τ-invariants alone. Then the remaining ones (which are called exceptional left cells) are characterized by the generalized τ-invariants together with the a-values.