Crystal of Affine sl^ _ℓ and Modular Branching Rules for Hecke Algebras of Type D_n

Let ℋ_q(B_n) and ℋ_q(D_n) denote the Hecke algebras of types B_n and D_n respectively, where q ≠ 1 is the Hecke parameter with quantum characteristic e. We prove that if D ^λ is a simple ℋ _q(B _2n)-module which splits as D+λ⊕D−λ upon restriction to ℋ_q(D _2n), then D+λ↓ℋq(D2n−1)≅D−λ↓ℋq(D2n−1) and D+λ↑ℋq(D2n+1)≅D−λ↑ℋq(D2n+1) . In particular, we get some multiplicity-free results for certain two-step modular branching rules. We also show that when e = 2ℓ > 2 the highest weight crystal of the irreducible sl^ _ℓ -module L(Λ₀) can be categorified using the simple ℋ_q (D _2n)-modules {D+λ∣λ=(λ(1),λ(2))⊢2n,Dλ↓ℋq(D2n)≅D+λ⊕D−λ,n∈ℕ} and certain two-step induction and restriction functors. Finally, a complete classification of all the simple blocks of ℋ_q(D_n) is also obtained.