INTEGRAL AND GRADED QUASI-HEREDITARY ALGEBRAS, II WITH APPLICATIONS TO REPRESENTATIONS OF GENERALIZED q-SCHUR ALGEBRAS AND ALGEBRAIC GROUPS

We consider a pair (a, A) consisting of a quasi-hereditary algebra A and a (positively) graded subalgebra a. We present conditions which guarantee that the algebra grA obtained by grading A by its radical filtration is quasi-hereditary and Koszul. In such cases, we also show that the standard and costandard modules for A have a structure as graded modules for a. These results are applied to obtain new information about the finite dimensional algebras (e. g., the q-Schur algebras) which arise from quantum enveloping algebras. In the final section, we consider an order à for a quasi-hereditary algebra AK , and consider conditions which guarantee that grà is an integral quasi-hereditary algebra. This section and many other parts of the paper have consequences for representations of algebraic groups in positive characteristic. Explicit positive characteristic analogues of the above results are given, with some restrictions.