Crystal of affine \widehat\mathfraksl_ℓ and Hecke algebras at a primitive 2ℓth root of unity

2019/05/17 by Lin, Huang, Hu, Jun
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1905.07333

Abstract

Let ℓ∈ℕ with ℓ>2 and I:=ℤ/2ℓℤ. In this paper we give a new realization of the crystal of affine \widehat\mathfraksl_ℓ 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 realized the Kashiwara operators for the crystal as the functors of taking socle of certain two-steps restriction and of taking head of certain two-steps induction. For any finite dimensional irreducible H_n-module M, we prove that the irreducible submodules of \rmres_H_n-2^H_nM which belong to \widehatB(∞) (Definition 6.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of affine \widehat\mathfraksl_ℓ and the affine Hecke algebras of type A at a primitive ℓth root of unity.

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