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

摘要

We study a relationship between Q-polynomial distance-regular graphs and the double affine Hecke algebra of type (C_1 v,C ~1). 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 xεC. 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 C-vector space W. The universal double affine Hecke algebra of type (C_1 v,C_1) is the C-algebra ?_q defined by generators {t_n±1}n=03 and relations (i) tntn-1=t_n-1tn=1; (ii) t_n+tn-1 is central; (iii) t_0t_1t_2t_3=q- ~(1/2). In this paper, we display an ?_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 Γ;t_3t_0+(t_3t _0)-~1 acts as the dual adjacency matrix of Γ with respect to C;t_1t_2+(t_1t_2)- 1 acts as the dual adjacency matrix of Γ with respect to x. To obtain our results we use the theory of Leonard systems.