  We introduce a generalized version of a q-Schur algebra (of parabolic type)
for arbitrary Hecke algebras over extended Weyl groups. We describe how the
decomposition matrix of a finite group with split BN-pair, with respect to a
non-describing prime, can be partially described by the decomposition matrices
of suitably chosen q-Schur algebras. We show that the investigated structures
occur naturally in finite groups of Lie type.
