We construct a commutative algebra Ax of difference operators in R p

Abstract. We construct a commutative algebra A z , generated by d algebraically independent q-difference operators acting on variables z 1 , z 2 , . . . , z d , which is diagonalized by the multivariable Askey-Wilson polynomials P n (z) considered by Gasper and Rahman (2005). Iterating Sears' 4 φ 3 transformation formula, we show that the polynomials P n (z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A n , generated by d algebraically independent difference operators acting on the discrete variables n 1 , n 2 , . . . , n d , which is also diagonalized by P n (z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.

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In 1971, Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra sl d+1 (C). Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d 2 + d + 1 terms. This, combined with the duality, establishes their bispectrality. We illustrate our results with several explicit examples.

We consider the generic quantum superintegrable system on the d-sphere with potential V (y) = d+1 k=1 b k y 2 k , where b k are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in d variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in [9]. The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik's multivariable Racah polynomials.