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

Abstract

Let ( W, S) be a Weyl group and H its associated Hecke algebra. Let A= Z[u,u −1] be the Laurent polynomial ring. Kazhdan and Lusztig [Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] introduced two A -bases { T w } w∈ W and { C w } w∈ W for the Hecke algebra H associated to W. Let Y w =∑ y⩽ w u l( w)− l( y) T y . Then { Y w } w∈ W is also an A -base for the Hecke algebra. In this paper we give an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of Y x 's in the Hecke algebra of type D n . In fact, this gives also an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of T x 's in the Hecke algebra of type D n . 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 n 2−3 n+3 in the Weyl group of type D n .