  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 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.
