  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
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 interpolation points that arise from regular
elements of finite order in $G$. The polynomials arise from the irreducible
characters of $G$ and the interpolation points 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.
