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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.

Generalized q -Schur algebras and modular representation theory of finite groups with split ( BN )-pairs