On Dipper–Mathas’s Morita equivalences
Volume 149 / 2017
Abstract
Dipper and Mathas have proved that every Ariki–Koike algebra (i.e., nondegenerate cyclotomic Hecke algebra of type $G(\ell ,1,n)$) is Morita equivalent to a direct sum of tensor products of some smaller Ariki–Koike algebras which have $q$-connected parameter sets. They proved this result by explicitly constructing a progenerator which induces this equivalence. In this paper we use the nondegenerate affine Hecke algebra $\mathcal {H}^{\rm aff}_n$ to derive Dipper–Mathas’s Morita equivalence as a consequence of an equivalence between the block $\mathcal {H}^{\rm aff}_n\mbox {-mod}[{\boldsymbol \gamma }]$ of the category of finite-dimensional modules over $\mathcal {H}^{\rm aff}_n$ and the block $\mathcal {H}^{\rm aff}_{n_1}\otimes \dots \otimes \mathcal {H}^{\rm aff}_{n_r}\mbox {-mod}[({\boldsymbol \gamma }^{(1)},\dots ,{\boldsymbol \gamma }^{(r)})]$ of the category of finite-dimensional modules over the parabolic subalgebra $\mathcal {H}^{\rm aff}_{n_1}\otimes \dots \otimes \mathcal {H}^{\rm aff}_{n_r}$ under certain conditions on ${\boldsymbol \gamma },{\boldsymbol \gamma }^{(1)},\ldots ,{\boldsymbol \gamma }^{(r)}$. Similar results for the degenerate versions of these algebras are also obtained.
