In [DJM] Richard Dipper, Gordon James and myself introduced the cyclotomic q-Schur algebras and showed that they are quasi-hereditary cellular algebras. By definition, a cyclotomic q-Schur algebra is a certain endomorphism algebra attached to an Ariki-Koike algebra in much the same way as the q-Schur algebra [DJ] is defined as an endomorphism algebra of a particular module for the Iwahori Hecke algebra of the symmetric group.

One of our motivations for defining the cyclotomic q-Schur algebras was to provide another tool for studying the Ariki-Koike algebras. Recently Gordon James and I [JM] used the cyclotomic q-Schur algebras to prove an analogue of the Jantzen Schaper theorem for the Ariki-Koike algebras. Most of the argument is devoted to first proving an analogue of Jantzen's sum formula for the Weyl modules of the cyclotomic q-Schur algebra. The result for the Ariki-Koike algebras is then deduced by a Schur functor argument.

As a special case of our results we obtained, for the first time, an analogue of the Jantzen Schaper theorem for Coxeter groups of type B.

In this talk I will describe these results, together with details of the proof and some applications.

[DJ] R. Dipper and G. James, The q-Schur algebra, Proc. London Math. Soc. (3), 59 (1989), 23-50.

[DJM] R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z., to appear.

[JM] G. James and A. Mathas, The cyclotomic Jantzen Schaper theorem, preprint.