Order Reversing Involution And Distinguished Involutions In Weyl Groups

This paper mainly includes three parts. In the first part, we give the method of computing the distinguished involution in a Weyl group. As an example, we calculate all the distinguished involutions of the Weyl group of type B4. The method mainly depends on left (right)*-operation and the computation of certain Kazhdan-Lusztig polynomials. In the paper [8], D. Kazhdan and G. Lusztig point out that "order reversing involution" x→w0x reverses left(right) and two-sided cells of a Weyl group. On the other hand, each left(right) cell contains exactly one distinguished involution. Thus we want to know how the order reversing involution acts on the distinguished involutions.Whether it is true that the resulted element is also a distinguished involution. So in part two we investigate carefully the relationship between order reversing involution and left(right)*-operation and double-sided *-operation. We obtain two interesting results. The first one is that left(right) *-operation is commutative with order reversing involution. The second is that double-sided *-operation is commutative with order reversing involution, provided that the degrees of the irrducible Weyl group(cf.[1]or[7]) are all even. In the third part, under the framework of Deodhar Lemma, we give detailed investigation on *(d*) for any distinguished involution d.