Matrix units and primitive idempotents for the Hecke algebra of type D_n

Let n∈Z^≥4 and H_q(D_n) be the semisimple Hecke algebra of type D_n with Hecke parameter q∈K^×. For each simple H_q(D_n)-module V, we use the Hecke generators of H_q(D_n) to construct explicitly a quasi-idempotent z_V (i.e., z_V²=c_Vz_V for some c_V∈K^×) which is defined over a natural integral form of H_q(D_n), such that e_V:=c_V⁻¹z_V is a primitive idempotent and e_VH_q(D_n)≅V as right H_q(D_n)-module. We use the seminormal bases of the Hecke algebra H_q(B_n) of type B_n to construct a complete set of pairwise orthogonal primitive idempotents of H_q(D_n), to obtain an explicit seminormal basis of H_q(D_n) as well as a new seminormal construction for each simple module over H_q(D_n). As byproducts, we discover some rational property of certain square-roots of quotients of γ-coefficients for H_q(B_n), which play a key role in the proof of the main results of the paper.