Morita equivalences for cyclotomic Hecke algebras of type B and D

Loïc Poulain d'Andecy (1) , Salim Rostam (2, 3, 4)

1 LMR - Laboratoire de Mathématiques de Reims
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 UR - Université de Rennes
4 ENS Rennes - École normale supérieure - Rennes

We give a Morita equivalence theorem for so-called cyclotomic quotients of affine Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike algebras. As a consequence, the representation theory of affine Hecke algebras of type B and D reduces to the study of their cyclotomic quotients with eigenvalues in a single orbit under multiplication by $q^2$ and inversion. The main step in the proof consists in a decomposition theorem for generalisations of quiver Hecke algebras that appeared recently in the study of affine Hecke algebras of type B and D. This theorem reduces the general situation of a disconnected quiver with involution to a simpler setting. To be able to treat types B and D at the same time we unify the different definitions of generalisations of quiver Hecke algebra for type B that exist in the literature.

Mots clés: Morita equivalence; Hecke algebra

Domaines: Théorie des représentations [math.RT]

Loïc Poulain d'Andecy, Salim Rostam. Morita equivalences for cyclotomic Hecke algebras of type B and D. Bulletin de la société mathématique de France, In press. ⟨hal-02104136v1⟩