On the decomposition numbers of the Hecke algebra of type D_n when n is even

Let n ≥ 4 be an even integer. Let K be a field with char K ≠ 2 and q an invertible element in K such that ∏_{i = 1}^{n - 1} (1 + qⁱ) ≠ 0. In this paper, we study the decomposition numbers over K of the Iwahori-Hecke algebra H_q (D_n) of type D_n. We obtain some equalities which relate its decomposition numbers with certain Schur elements and the decomposition numbers of various Iwahori-Hecke algebras of type A with the same parameter q. When char K = 0, this completely determine all of its decomposition numbers. The main tools we used are the Morita equivalence theorem established in [J. Hu, A Morita equivalence theorem for Hecke algebra H_q (D_n) when n is even, Manuscripta Math. 108 (2002) 409-430] and certain twining character formulae of Weyl modules over a tensor product of two q-Schur algebras.