Length-generating functions for twisted involutions of Coxeter groups

For a Coxeter system $(W,S)$ of finite rank and an involutive automorphism $\ast :W\to W$ which preserves $S$, we prove that the length-generating function ${\sum }_{z\in {\mathbf{I}}_{W,\ast }}{q}^{\mathrm{\ell }(z)}$ of the set of $\ast$-twisted involutions is rational. We derive recurrence relations analogous to the well-known recurrence relations expressing the growth series $W(q):={\sum }_{w\in W}{q}^{\mathrm{\ell }(w)}$ in terms of the growth series $W_I(q)$ of the parabolic subgroups of $W$. Our result extends to any subset $\mathcal{C}$ of $\ast$-twisted involutions that is closed under $\ast$-twisted conjugation. We use these recurrence relations to give an alternative proof of a power-series identity for twisted involutions due to Lusztig.