Extending Hecke Endomorphism Algebras at Roots of Unity

Hecke endomorphism algebras are endomorphism algebras over a Hecke algebra associated to a finite Weyl group W of certain q-permutation modules, the “tensor spaces.” Such a space may be defined for any W in terms of a direct sum of certain cyclic modules associated to parabolic subgroups. The associated algebras have important applications to the representations of finite groups of Lie type. In [6], it is proved that these algebras can be stratified by means of a filtration defined in terms of the subsets of the Coxeter generators. It was conjectured that by enlarging the “tensor space” the new resulting endomorphism algebra has a finer “standard” stratification in terms of left cells of the Coxeter group, with associated “strata” corresponding to two-sided cells of W . Using the work [8] on a rational double affine Hecke algebras (RDAHAs)—also known as rational Cherednik algebras—at a key point, we will prove the conjecture in the characteristic zero case at an eth root of unity, e 6= 2. We further prove that each of the new Hecke endomorphism algebras constructed in the paper is quasi-hereditary and that its representation category is equivalent, after a base change, to the category O associated to a corresponding RDAHA. We do not treat the e = 2 case, but expect the conjecture to be true there also (possibly not giving quasi-hereditary algebras, in general).