R. Dipper et J. Gruber, Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs, J REIN MATH, 511, 1999, pp. 145-191
We introduce a generalized version of a q-Schur algebra (of parabolic type)
for arbitrary Hecke algebras over extended Weyl groups. We describe how th
e decomposition matrix of a finite group with split BN-pair, with respect t
o a non-describing prime, can be partially described by the decomposition m
atrices of suitably chosen q-Schur algebras. We show that the investigated
structures occur naturally in finite groups of Lie type.