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

Let $\mathrm{\Gamma }$ denote a $Q$-polynomial distance-regular graph with diameter $D \geq 3$andstandardmodule$V$. Recently Ito and Terwilliger introduced four direct sum decompositions of $V$; we call these the $(\mu ,\nu )$–{\it split decompositions} of $V$, where $\mu, \nu \in \lbrace \downarrow, \uparrow \rbrace$\mathrm{.}Inthispaperweshowthatthe($\downarrow,\downarrow$)–split decomposition and the ($\uparrow ,\uparrow$)–split decomposition are dual with respect 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 the standard Hermitian form on $V$.