Algebra of Dunkl Laplace-Runge-Lenz vector

We introduce the Dunki version of the Laplace-Runge-Lenz vector associated with a finite Coxeter group W acting geometrically in R-N and with a multiplicity function g. This vector generalizes the usual Laplace-Runge-Lenz vector and its components commute with the Dunkl-Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential gamma/r. We study the resulting symmetry algebra R-g,R-gamma ( W) and show that it has the Poincard-Birkhoff-Witt property. In the absence of a Coulomb potential, this symmetry algebra R-g,R-0 (W) is a subalgebra of the rational Cherednik algebra H-g(W). We show that a central quotient of the algebra R-g,R-gamma (W) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra H-g(so(N+1)) (W). This gives an interpretation of the algebra H-g(so(N)+(1)) (W) as the hidden symmetry algebra of the Dunkl-Coulomb problem in R-N. By specialising R-g,R-gamma(W) to g = 0, we recover a quotient of the universal enveloping algebra U (so(N+1)) as the hidden symmetry algebra of the Coulomb problem in R-N. We also apply the Dunki Laplace-Runge-Lenz vector to establish the maximal superintegrability of the generalised Calogero-Moser systems.