Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W. The length polynomial LW,X(t) is defined by LW,X(t)=∑x∈Xtℓ(x), where ℓ is the length function on W. If X={x∈W:x2=1} then we call LW,X(t) the involution length polynomial of W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W. In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type Bn and Dn. Moreover, we give a counterexample to a unimodality conjecture stated in [11].