Crystal of affine type (A)over-capl-1 and Hecke algebras at aprimitive 2lth root of unity

Let l is an element of Nwith l > 1. In this paper we give a new realization of the crystal of affine type A(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 RepI(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 is an element of RepI(H-n), we prove that the simple submodules of res(Hn)(Hn) - 2M which belong to B(infinity)( Definition5.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of sl(l) and the affine Hecke algebras of type Aat a primitive lth root of unity. (C) 2021 Elsevier Inc. All rights reserved.