Left Cells In The Affine Weyl Groups Of Type E

In this paper, by using Shi's algorithm we describe the left cells in two-sided cells of the affine Weyl group E6 with a-value < 11 and construct the corresponding left cells graphs. We also describe the left cells in the two-sided of the affine Weyl groups E7 and E$ with a-value 4.Then we work out the distinguished involutions of left cells in the two-sided cells of the arBne Weyl groups of type E with a-value 4, and we prove that these left cells are all left-connected, which verify a conjecture of Lusztig in this case.