Parabolic Temperley–Lieb modules and polynomials

We define and study, for any Coxeter system (W,S)(W,S), modules over its Temperley–Lieb algebra, two for each quotient WJWJ, which have generators indexed by the fully commutative elements of WJWJ. Our results are new even in type A   and include, for J=∅J=∅, those obtained in [16] and [17].