Bar operators for quasiparabolic conjugacy classes in a Coxeter group

TL;DR

This paper develops a theory of bar operators and Kazhdan-Lusztig bases for quasiparabolic conjugacy classes in Coxeter groups, extending classical representation theory tools to new algebraic structures.

Contribution

It introduces a natural definition of quasiparabolic bar operators and proves their existence for certain twisted conjugacy classes, advancing the understanding of quasiparabolic modules.

Findings

Existence of quasiparabolic bar operators for twisted conjugacy classes.

Development of quasiparabolic Kazhdan-Lusztig bases theory.

Classification results for quasiparabolic conjugacy classes.

Abstract

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.