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

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Let n≥ 4 be an even integer. Let K be a field with K≠ 2 and q an invertible element in K such that ∏ᵢ₌₁ⁿ⁻¹(1+qⁱ)≠ 0. In this paper, we study the decomposition numbers over K of the Iwahori--Hecke algebra (Dₙ) of type Dₙ. 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 K=0, this completely determine all of its decomposition numbers. The main tools we used are the Morita equivalence theorem established in Hu1 and certain twining character formulae of Weyl modules over a tensor product of two q-Schur algebras.