Hecke algebras of type D_n at roots of unity

This paper deals with representations of Hecke algebras of type D_n(i.e., H_q(D_n)) over any fieldKand with any parameter q. Our method is to study the restrictions (to H_q(D_n)) of the Specht moduleS̃^λ and simple moduleD̃^λ of H_q,1(B_n). We prove that S̃^λ|H_q(D_n) ≅ S̃^λ̂|H_q(D_n), where λ̂ = (λ⁽²⁾, λ⁽¹⁾). When λ⁽¹⁾ = λ⁽²⁾, we show that S̃^λ is a direct sum of its two H_q(D_n)-submodules with the same dimensions. Via this we give a crude presentations of all the simple modules of H_q(D_n). When H_q(D_n) is semisimple, we give a complete set of pairwise non-isomorphic simple modules. Finally, we also give an attempt of using "Murphy basis" philosophy and construct a basis for H_q(D_n) when n is odd.