Crystal of affine type Aˆ_ℓ−1 and Hecke algebras at a primitive 2ℓth root of unity

Let ℓ∈N with ℓ>1. In this paper we give a new realization of the crystal of affine type Aˆ_ℓ−1 using the modular representation theory of the affine Hecke algebras H_n of type A and their level two cyclotomic quotients with Hecke parameter being a primitive 2ℓth root of unity. We construct “hat” versions of i-induction and i-restriction functors on the category Rep_I(H_n) of finite dimensional integral modules over H_n, which induce Kashiwara operators on a suitable subgroup of the Grothendieck groups of Rep_I(H_n). For any simple module M∈Rep_I(H_n), we prove that the simple submodules of res_Hn−2^H_nM which belong to Bˆ(∞) (Definition 5.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of slˆ_ℓ and the affine Hecke algebras of type A at a primitive ℓth root of unity.