Bar operators for quasiparabolic conjugacy classes in a Coxeter group

The action of a Coxeter group $W$ on the set of left cosets of a standard parabolic subgroup deforms to define a module $\mathcal{M}^J$ of the group's Iwahori-Hecke algebra $\mathcal{H}$ with a particularly simple form. Rains and Vazirani have introduced the notion of a quasiparabolic set to characterize $W$-sets for which analogous deformations exist; a motivating example is the conjugacy class of fixed point free involutions in the symmetric group. Deodhar has shown that the module $\mathcal{M}^J$ possesses a certain antilinear involution, called the bar operator, and a certain basis invariant under this involution, which generalizes the Kazhdan-Lusztig basis of $\mathcal{H}$. The well-known significance of this basis in representation theory makes it natural to seek to extend Deodhar's results to the quasiparabolic setting. In general, the obstruction to finding such an extension is the existence of an appropriate quasiparabolic analogue of the "bar operator." In this paper, we consider the most natural definition of a quasiparabolic bar operator, and develop a theory of "quasiparabolic Kazhdan-Lusztig bases" under the hypothesis that such a bar operator exists. Giving content to this theory, we prove that a bar operator in the desired sense does exist for quasiparabolic $W$-sets given by twisted conjugacy classes of twisted involutions. Finally, we prove several results classifying the quasiparabolic conjugacy classes in a Coxeter group.