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We study the blocks for Ariki-Koike algebras using a general notion of core for $l$-partitions. We interpret the action of the affine symmetric group on the blocks in the context of level rank duality and study the orbits under this action.

The complexity of a block of a symmetric algebra can be measured by the notion of defect, a numerical datum associated to each of the simple modules contained in the block. Geck showed that the defect is a block invariant for Iwahori-Hecke algebras of finite Coxeter groups in the equal parameter case, and speculated that a similar result should hold in the unequal parameter case. We conjecture that the defect is a block invariant for all cyclotomic Hecke algebras associated with complex reflection groups, and we prove it for the groups of type $G(l,p,n)$ and for the exceptional types for which the blocks are known. In particular, for the groups $G(l,1,n)$, we show that the defect corresponds to the notion of weight in the sense of Fayers, for which we thus obtain a new way of computation. We also prove that the defect is a block invariant for cyclotomic Yokonuma-Hecke algebras., Comment: 25 pages