A group commutator involving the last distance matrix and dual distance matrix of a $Q$-polynomial distance-regular graph

Let $\mathrm{\Gamma }$ denote the Hamming graph $H(D,r)$ with $r\ge 3$. Consider the distance matrices $\{A_i{\}}_{i=0}^D$ of $\mathrm{\Gamma }$. Fix a vertex $x$ of $\mathrm{\Gamma }$, and consider the dual distance matrices $\{A_i^{\ast }{\}}_{i=0}^D$ of $\mathrm{\Gamma }$ with respect to $x$. We investigate the group commutator $A_D^{-1}{A}_D^{\ast -1}{A}_D{A}_D^{\ast }$. We show that this matrix is diagonalizable. We compute its eigenvalues and their eigenspaces. Let $T$ denote the subconstituent algebra of $\mathrm{\Gamma }$ with respect to $x$. We describe the action of $A_D^{-1}{A}_D^{\ast -1}{A}_D{A}_D^{\ast }$ on each irreducible $T$-module.