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 (C1∨,C1). Let Γ denote a Q-polynomial distance-regular graph with vertex set X. We assume that Γ has q-Racah type and contains a Delsarte clique C  . Fix a vertex x∈Cx∈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 CC-vector space W.The universal double affine Hecke algebra of type (C1∨,C1) is the CC-algebra Hˆq defined by generators {tn±1}n=03 and relations (i) tntn−1=tn−1tn=1; (ii) tn+tn−1 is central; (iii) t0t1t2t3=q−1/2t0t1t2t3=q−1/2. In this paper, we display an Hˆq-module structure for W. For this module and up to affine transformation,
• t0t1+(t0t1)−1t0t1+(t0t1)−1 acts as the adjacency matrix of Γ;
• t3t0+(t3t0)−1t3t0+(t3t0)−1 acts as the dual adjacency matrix of Γ with respect to C;
• t1t2+(t1t2)−1t1t2+(t1t2)−1 acts as the dual adjacency matrix of Γ with respect to x. To obtain our results we use the theory of Leonard systems.