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

Abstract: We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $\Gamma$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$. We partition $X$ according to the path-length distance to both $x$ 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 ${\bf W}$. The universal double affine Hecke algebra of type $(C^{\vee}_1,C_1)$ is the $\mathbb{C}$-algebra $\hat{H}q$ defined by generators ${t^{\pm1}_n}^3{n=0}$ and relations (i) $t_nt_n^{-1}=t_n^{-1}t_n=1$; (ii) $t_n+t_n^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1/2}$. In this paper, we display an $\hat{H}_q$-module structure for ${\bf W}$. For this module and up to affine transformation, (i) $t_0t_1+(t_0t_1)^{-1}$ acts as the adjacency matrix of $\Gamma$; (ii) $t_3t_0+(t_3t_0)^{-1}$ acts as the dual adjacency matrix of $\Gamma$ with respect to $C$; (iii) $t_1t_2+(t_1t_2)^{-1}$ acts as the dual adjacency matrix of $\Gamma$ with respect to $x$. To obtain our results we use the theory of Leonard systems.