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Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 3$ and standard module $V$. Recently Ito and Terwilliger introduced fourdirect sum decompositions of $V$; we call these the $(\mu,\nu)$--{\it splitdecompositions} of $V$, where $\mu, \nu \in \lbrace \downarrow, \uparrow\rbrace$. In this paper we show that the ($\downarrow,\downarrow$)--splitdecomposition and the ($\uparrow,\uparrow$)--split decomposition are dual withrespect to the standard Hermitian form on $V$. We also show that the($\downarrow,\uparrow$)--split decomposition and the($\uparrow,\downarrow$)--split decomposition are dual with respect to thestandard Hermitian form on $V$.

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