Filtrations on the Mackey decomposition for cyclotomic Hecke algebras

In [3], Grojnowski defines functors ei :RepHλn→RepHλn−1 and fi :RepHλn→RepHλn+1 shows that in the Grothendieck group K(RepHλn) [ei,fi]M=hi(M)M, where hi(M) is an integer depending only on λ and the central character of M. In this paper, we show that the above relation makes sense when M is an irreducible Hλn-module before passing to the Grothendieck group. That is, in the case hi(M) 0 eifi fieiM hi(M)M and in the case hi(M) 0 where for k 0, kM denotes the direct sum j=1kM.