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

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