Lusztigʹs isomorphism theorem for cellular algebras

In this paper, we first introduce a notion of semisimple system with parameters, then we establish Lusztigʹs isomorphism theorem for any cellular semisimple system with parameters. As an application, we obtain Lusztigʹs isomorphism theorem for Ariki–Koike algebras, cyclotomic q-Schur algebras and Birman–Murakami–Wenzl algebras. Second, using the results for certain Ariki–Koike algebras, we prove an analogue of Lusztigʹs isomorphism theorem for the cyclotomic Hecke algebras of type G(p,p,n) (which are not known to be cellular in general). These generalize earlier results of [G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981) 490–498.] on such isomorphisms for Iwahori–Hecke algebras associated to finite Weyl groups.