The universal DAHA of type $(C_1^\vee,C_1)$ and Leonard pairs of $q$-Racah type

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\mathbb F$ denote an algebraically closed field, and fix a nonzero $q \in \mathbb F$ that is not a root of unity. The universal double affine Hecke algebra (DAHA) $\hat{H}_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t_i^{\pm 1}\rbrace_{i=0}^3$ and relations (i) $t_it_i^{-1}=t_i^{-1}t_i=1$; (ii) $t_i+t_i^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1}$. We consider the elements $X=t_3t_0$ and $Y=t_0t_1$ of $\hat{H}_q$. Let $\mathcal V$ denote a finite-dimensional irreducible $\hat{H}_q$-module on which each of $X$, $Y$ is diagonalizable and $t_0$ has two distinct eigenvalues. Then $\mathcal V$ is a direct sum of the two eigenspaces of $t_0$. We show that the pair $X+X^{-1}$, $Y+Y^{-1}$ acts on each eigenspace as a Leonard pair, and each of these Leonard pairs falls into a class said to have $q$-Racah type. Thus from $\mathcal V$ we obtain a pair of Leonard pairs of $q$-Racah type. It is known that a Leonard pair of $q$-Racah type is determined up to isomorphism by a parameter sequence $(a,b,c,d)$ called its Huang data. Given a pair of Leonard pairs of $q$-Racah type, we find necessary and sufficient conditions on their Huang data for that pair to come from the above construction.