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

Let $n\geq 4$ be an even integer. Let $K$ be a field with $\cha K\neq 2$ and $q$ an invertible element in $K$ such that $\prod_{i=1}^{n-1}(1+q^i)\neq 0$. In this paper, we study the decomposition numbers over $K$ of the Iwahori--Hecke algebra $\HH_q(D_n)$ 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 $\cha K=0$, this completely determine all of its decomposition numbers. The main tools we used are the Morita equivalence theorem established in \cite{Hu1} and certain twining character formulae of Weyl modules over a tensor product of two $q$-Schur algebras.