The distinguished involutions with a-value n2−3n+3 in the Weyl group of type Dn

Let (W,S) be a Weyl group and H its associated Hecke algebra. Let be the Laurent polynomial ring. Kazhdan and Lusztig [Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (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 -base for the Hecke algebra. In this paper we give an explicit expression for certain Kazhdan–Lusztig basis elements Cw as -linear combination of Yxʹs in the Hecke algebra of type Dn. In fact, this gives also an explicit expression for certain Kazhdan–Lusztig basis elements Cw as -linear combination of Txʹs in the Hecke algebra of type Dn. Thus we describe also explicitly the Kazhdan–Lusztig polynomials for certain elements of the Weyl group. We study also the joint relation among some elements in W and some distinguished involutions with a-value n2−3n+3 in the Weyl group of type Dn.