  We solve two open problems in Coxeter-Catalan combinatorics. First, we
introduce a family of rational noncrossing objects for any finite Coxeter
group, using the combinatorics of distinguished subwords. Second, we give a
type-uniform proof that these noncrossing Catalan objects are counted by the
rational Coxeter-Catalan number, using the character theory of the associated
Hecke algebra and the properties of Lusztig's exotic Fourier transform. We
solve the same problems for rational noncrossing parking objects.
