The Distinguished Involutions Of The Affine Weyl Group Of Type (?)

In the paper, we study the distinguished involutions of left cells in the affine Weyl group E6. We use four methods to find one distinguished involution d such that the left cellгd containing d occurs as a vertex of ML in. each left-cell graph ML, Then by Thoerem 1.7.2.5 we apply two-sided star operations in matlab to find the set R of distinguished in-volutions such that{гd'|d'∈R} form the vertex set of ML. We draw the distinguished involutions graph corresponding to ML.Finally, We prove that all the left cells in W(5) and W(6) are left-connected, verifying a conjecture of Lusztig in our case.