The displacement and split decompositions for a $Q$-polynomial distance-regular graph

Let $\mathrm{\Gamma }$ denote a $Q$-polynomial distance-regular graph with diameter at least three and standard module $V$. We introduce two direct sum decompositions of $V$. We call these the displacement decomposition for $\mathrm{\Gamma }$ and the split decomposition for $\mathrm{\Gamma }$. We describe how these decompositions are related.