Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs

In his famous theorem (1982), Douglas Leonard characterized the q-Racah polynomials and their relatives in the Askey scheme from the duality property of Q-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the q-Racah polynomials in the above situation. Let Γ denote a Q-polynomial distance-regular graph that contains a Delsarte clique C. Assume that Γ has q-Racah type. Fix a vertex x∈C. We partition the vertex set of Γ according to the path-length distance to both x and C. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra Hˆq of type (C1∨,C1). From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the Hˆq-module and the theory of Leonard systems. Changing Hˆq by Hˆq−1 we show how our Laurent polynomials are related to the nonsymmetric Askey-Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric q-Racah polynomials.