Geodesic
Deformations of permutation representations of Coxeter groups.
Journal of Algebraic Combinatorics, Tome 37 (2013) no. 3, pp. 455-502.

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Summary: The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over $\Bbb $Z[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of "quasiparabolic" subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.
Keywords: Coxeter group, permutation representation, Hecke algebra, Bruhat order 
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     title = {Deformations of permutation representations of {Coxeter} groups.},
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Rains, Eric M.; Vazirani, Monica J. Deformations of permutation representations of Coxeter groups.. Journal of Algebraic Combinatorics, Tome 37 (2013) no. 3, pp. 455-502. http://geodesic.mathdoc.fr/item/JAC_2013__37_3_a4/