A duality between pairs of split decompositions for a QQ-polynomial distance-regular graph

Let ΓΓ denote a QQ-polynomial distance-regular graph with diameter D≥3D≥3 and standard module VV. Recently, Ito and Terwilliger introduced four direct sum decompositions of VV; we call these the (μ,ν)(μ,ν)-split decompositions   of VV, where μ,ν∈{↓,↑}μ,ν∈{↓,↑}. In this paper we show that the (↓,↓↓,↓)-split decomposition and the (↑,↑↑,↑)-split decomposition are dual with respect to the standard Hermitian form on VV. We also show that the (↓,↑↓,↑)-split decomposition and the (↑,↓↑,↓)-split decomposition are dual with respect to the standard Hermitian form on VV.