Kazhdan-Lusztig cells and decomposition numbers

Abstract

We consider a generic Iwahori–Hecke algebra H H associated with a finite Weyl group. Any specialization of H 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 H to the asymptotic algebra  J 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 1 case.