Q-polynomial distance-regular graphs and the DAHA of type (C^V₁, C₁)

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 W. The universal DAHA of type (C^{\vee}_1, C_1) is the \mathbb{C}-algebra \hat{H}_q defined by generators {t_n^{\pm1}}^3_{n=0} and relations (i) t_nt^{-1}_n = t^{-1}_nt_n = 1; (ii) t_n+t_n^{-1} is central; (iii) t_0t_1t_2t_3 = q^{-1/2}. We display an \hat{H}_q-module structure for W. For this module and up to affine transformation,

• t_0t_1+(t_0t_1)^{-1} acts as the adjacency matrix of Gamma;
• t_3t_0+(t_3t_0)^{-1} acts as the dual adjacency matrix of Gamma with respect to C;
• t_1t_2+(t_1t_2)^{-1} acts as the dual adjacency matrix of Gamma with respect to x.