Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t)={\sum }_{x\in X}{t}^{?(x)}$, where ? is the length function on W. If $X=\{x\in W:x^2=1\}$ then we call $L_{W,X}(t)$ 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 $B_n$ and $D_n$. Moreover, we give a counterexample to a unimodality conjecture stated in [11].