Two-sided Cells With A-value≤6 In The Affine Weyl Group Of Type (?)₆

Let W= (W, S) be a Coxeter group. D.Kazhdan and G.Lusztig defined the concepts of cell decomposition in the affine Weyl group W,which can be decomposed into the union of the disjoint two-sided cells, while each two-sided cell can be decomposed into the union of the disjoint left cells, and every left cell has the unique distinguished involution. If all the distinguished involutions can be found, then the problem of decomposition of the cells of W will be solved. In this paper, we obtain the distinguished involutions by seeking the primitive elements and by right star operation and left star operation on W, Therefore, we obtain two-sided cells with a-value< 6 in the affine Weyl group of type D6.We get the following result:When a-value is 4,we get 2 two-sided cells which contain 60 and 118 left cells;When a-value is 5,we get 1 two-sided cell which contain 155 left cells;When a-value is 6,we get 3 two-sided cells which contain 120,286 and 504 left cells.