The Distinguished Involutions Of The Affine Weyl Groups Of Type (?)₄,(?)₄,(?)₄ And (?)₄

Let W be the affine Weyl group of type B4,C4,D4 or F4. In this paper, we mainly study the left cells in W, and describe all the distinguished involutions of the affine Weyl group of type B4,C4 and D4. Because of the complexity of the computation of the Kazhdan-Lusztig polynomials, we find the mass of the distinguished involutions of the affine Weyl group of type F4 except the distinguished involuntions of few left cells that the function a is equal to sixteen. By means of the left cell graph, we only need to find one single distinguished involution. Then applying thoerem 1.7.2.7 and corollary 1.7.2.8, we can find all the distinguished involutions of the left cells of this left cell graph by programming matlab. In the process of searching the distinguished involution, we will mainly use the following three methods:Firstly, in the left cell graph M of the two-sided cell that the function a is integer, if the graph M appears the vertex labeled J, where J(?)S, l(wJ)=t, we will find all the distinguished involutions of the left cells of this graph. Secondly, we suppose that the graph M1 and graph M2 are isomorphic. If the distinguished involution of the vertex of graph M1 has been found, we will obtain the distinguished involution of the corresponding vertex of graph M2 by applying the automorphism of graph. Thirdly, we will search the involutions satisfying certain conditions by using mathematical software Coxeter versions3.0 and matlab, and then prove whether it is a distinguished involution by computing the Kazhdan-Lusztig polynomials. Finally, we will give out the distinguished involution graph with corresponding the left cell graph in this paper. And we will clearly show the distinguished involution represented by the hollow vertex of each the distinguished involution graph in the text.