On certain distinguished involutions in the Weyl group of type Dn

Let (W,S) be a Weyl group. Let be the Laurent polynomial ring in an indeterminate u. Kazhdan and Lusztig [Invent. Math. 33 (1979) 165–184] introduced two -bases {Tw}w W and {Cw}w W for the Hecke algebra H associated to W. Let Yw=∑y wul(w)−l(y)Ty. Then {Yw}w W is also an -basis for the Hecke algebra. In this paper we assume W of type Dn and we express certain Kazhdan–Lusztig basis elements Cw as -linear combination of Yxʹs. This in turn gives an explicit expression for certain Kazhdan–Lusztig basis elements Cw as -linear combination of Txʹs. Thus we describe explicitly the Kazhdan–Lusztig polynomials for certain pairs of elements of W. We also study the joint relation among some elements in W. In particular, we find certain distinguished involutions in the two-sided cell Ωt of W with a-value for 1 2t n and n even ( for n odd), where the two-sided cell Ωt does not contain the longest element (w0)J in subgroup WJ of W for any J S.