Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups

The paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of n variables based on the root lattices of compact simple Lie groups G of any type and of any rank n. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from A-type lattices, yield Gaussian cubature formulae for each simple Lie group G based on nodes (interpolation points) that arise from regular elements of finite order in G. The polynomials arise from the irreducible characters of G and the nodes as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections.