Stratifying endomorphism algebras using exact categories

This paper constructs enlargements of Hecke algebras over Z[t,t−1] to certain standardly stratified algebras. The latter are obtained as endomorphism algebras of modules with dual left cell module filtrations in the sense of Kazhdan-Lusztig. A novel feature of the proofs is the use of suitably chosen exact categories to avoid difficult Ext1-vanishing conditions.