The Left Cells With A=5, 6 In Affine Weyl Group (?)₈

In the paper,we study certain left cells in the affine Weyl group (?)₈.Moreprecisely,we study the left cells in the two-sided cell W₅ with a(W₅)=5 and oneof two two-sided cellsΩwith a(Ω)=6 (written W₆¹).We find a representative setof left cells in W₅ and W₆¹,and draw the corresponding left cells graphs.We provethat all the left cells in W₅ and W₆¹ are left-connected,verifying a conjecture ofLusztig in our case.Finally,we find all the distinguished involutions of (?)₈ in W₅and W₆¹.