The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard triples

Assume that $\mathbb{F}$ is an algebraically closed field and $q$ is a nonzero scalar in $\mathbb{F}$ that is not a root of unity. The universal Askey–Wilson algebra ${\mathrm{△}}_q$ is a unital associative $\mathbb{F}$-algebra generated by $A,B,C$ and the relations state that each of $$A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}},B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}},C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$$ is central in ${\mathrm{△}}_q$. The universal DAHA ${\mathfrak{H}}_q$ of type $(C_1^{\vee },C_1)$ is a unital associative $\mathbb F$-algebrageneratedby${t_i^{\pm 1}}_{i=0}^3$ and the relations state that \begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \ \hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \ t_0t_1t_2t_3=q^{-1}. \end{gather*} It was given an $\mathbb{F}$-algebra homomorphism ${\mathrm{△}}_q\to {\mathfrak{H}}_q$ that sends \begin{eqnarray*} A &\mapsto & t_1 t_0+(t_1 t_0)^{-1}, \ B &\mapsto & t_3 t_0+(t_3 t_0)^{-1}, \ C &\mapsto & t_2 t_0+(t_2 t_0)^{-1}. \end{eqnarray*} Therefore any $\mathfrak H_q$-modulecanbeconsideredasa$\triangle_q$-module\mathrm{.}Let$V$ denote a finite-dimensional irreducible ${\mathfrak{H}}_q$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the ${\mathrm{△}}_q$-module $V$.