Some matrices associated with the split decomposition for a Q-polynomial distance-regular graph

Understand

We consider a $Q$-polynomial distance-regular graph $\Gamma$ with vertex set $X$ and diameter $D \geq 3$.

• For $\mu, \nu \in \lbrace \downarrow, \uparrow \rbrace$ we define a direct sum decomposition of the standard module $V=\C X$, called the $(\mu,\nu)$--split decomposition.
• For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents.
• Now fix $b,\beta \in \mathbb C$ such that $b \neq 1$ and assume $\Gamma$ has classical parameters $(D,b,\alpha,\beta)$ with $\alpha = b-1$.