A classification of commutative parabolic Hecke algebras

Let (W,S) be a Coxeter system with I⊆S such that the parabolic subgroup WI is finite. Associated to this data there is a Hecke algebra H and a parabolic Hecke algebra HI=1IH1I (over a ring Z[qs]s∈S). We give a complete classification of the commutative parabolic Hecke algebras across all Coxeter types.