The Jones-Wenzl idempotent of a generalized Temperley-Lieb algebra

We define an idempotent of the Temperley-Lieb algebra of any finite Coxeter system, which reduces, in type A, to the known Jones-Wenzl idempotent. We give, for any pair of parabolic subgroups, a recursive formula generalizing the well-known one. This approach gives a wide class of recursive formulas for the classical Jones-Wenzl idempotent. We also compute explicitly the coefficient corresponding to the maximal element of any minuscule quotient, when the idempotent is expressed in the basis of fully commutative elements.