Crystal bases and simple modules for Hecke algebra of type D_n

Let H_ℂ(B_n) (respectively H_ℂ(D_n)) be the Hecke algebra of type B_n (respectively of type D_n) over the complex numbers field ℂ. Let ζ be a primitive 2ℓth root of unity in ℂ. For any Kleshchev bipartition (with respect to (ζ, 1, -1 ) λ = (λ⁽¹⁾, λ⁽²⁾) of n, let D̃^λ be the corresponding irreducible H_ℂ(B_n)-module. In the present paper we explicitly determine which D̃^λ split and which D̃^λ remains irreducible when restricts to H_ℂ(D_n). This yields a complete classification of all the simple modules for Hecke algebra H_ℂ(D_n). Our proof makes use of the crystal bases theory for the Fock representation of the quantum affine algebra U₁(sl_2ℓ) and deep result of Ariki's proof of LLT's conjecture [J. Math. Kyoto Univ. 36 (1996) 789-808].