In this project the principal investigator will continue his work on orthogonal polynomials and integrals associated to Coxeter groups and root systems by means of a parametrized commutative algebra of differential-difference operators. In particular, he will look at the intertwining operator which relates these algebras for different parameter values. This operator is conjectured to be a new type of fractional integral operator supported by a subset of the group algebra. There is a relationship to Poisson kernels that could lead to explicit integral forms. The principal investigator is analyzing orthogonal polynomials that are associated to Coxeter groups and their root systems, in order to study questions involving differential- difference operators, kernels of integral operators, and special functions associated with the Laplace operator. Orthogonal polynomials arise in an essential way in a number of different areas, such as numerical analysis, approximation theory and differential equations. In this project the principal investigator will use the algebraic structure associated with these polynomials to extend our knowledge of the symmetry properties of the polynomials.