$Q$-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C_1^{\vee },C_1)$. Let $\mathrm{\Gamma }$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $\mathrm{\Gamma }$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$\mathrm{.}Wepartition$X$accordingtothepath-lengthdistancetoboth$x$and$C$. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a $\mathbb{C}$-vector space $\mathbf{W}$. The universal double affine Hecke algebra of type $(C_1^{\vee },C_1)$ is the $\mathbb{C}$-algebra ${\widehat{H}}_q$ defined by generators $\{t_n^{\pm 1}{\}}_{n=0}^3$ and relations (i) $t_n{t}_n^{-1}=t_n^{-1}{t}_n=1$; (ii) $t_n+t_n^{-1}$ is central; (iii) $t_0{t}_1{t}_2{t}_3=q^{-1\mathrm{/}2}$. In this paper, we display an ${\widehat{H}}_q$-module structure for $\mathbf{W}$. For this module and up to affine transformation, (i) $t_0{t}_1+(t_0{t}_1{)}^{-1}$ acts as the adjacency matrix of $\mathrm{\Gamma }$; (ii) $t_3{t}_0+(t_3{t}_0{)}^{-1}$ acts as the dual adjacency matrix of $\mathrm{\Gamma }$ with respect to $C$; (iii) $t_1{t}_2+(t_1{t}_2{)}^{-1}$ acts as the dual adjacency matrix of $\mathrm{\Gamma }$ with respect to $x$. To obtain our results we use the theory of Leonard systems.