On certain distinguished involutions in the Weyl group of type Dn

Abstract

Let ( W, S) be a Weyl group. Let A= Z[u,u −1] be the Laurent polynomial ring in an indeterminate u. Kazhdan and Lusztig [Invent. Math. 33 (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 -basis for the Hecke algebra. In this paper we assume W of type D n and we express certain Kazhdan–Lusztig basis elements C w as A -linear combination of Y x 's. This in turn gives an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of T x 's. 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 Ω t of W with a-value 1 2 (n 2−n+4t 2−2t) for 1⩽2 t⩽ n and n even ( 1 2 (n 2−n+4t 2+2t) for n odd), where the two-sided cell Ω t does not contain the longest element ( w 0) J in subgroup W J of W for any J⊂ S.