Kazhdan–lusztig Cells and Decomposition Numbers

We consider a generic Iwahori–Hecke algebra H associated with a finite Weyl group. Any specialization of H gives rise to a corresponding decomposition matrix, and we show that the problem of computing that matrix can be interpreted in terms of Lusztig’s map from H to the asymptotic algebra J . This interpretation allows us to prove that the decomposition matrices always have a lower uni-triangular shape; moreover, we determine these matrices explicitly in the so-called defect 1 case.