The generic quantum superintegrable system on the sphere and Racah operators

We consider the generic quantum superintegrable system on the d-sphere with potential \(V(y)=\sum _{k=1}^{d+1}\frac{b_k}{y_k^2}\), 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 Geronimo and Iliev (Constr Approx 31(3):417–457, 2010). The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik’s multivariable Racah polynomials.