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The paper contains a generalization of known properties of Chebyshevpolynomials of the second kind in one variable to polynomials of $n$ variablesbased on the root lattices of compact simple Lie groups $G$ of any type and ofrank $n$. The results, inspired by work of H. Li and Y. Xu where they derivedcubature formulae from $A$-type lattices, yield Gaussian cubature formulae foreach simple Lie group $G$ based on interpolation points that arise from regularelements of finite order in $G$. The polynomials arise from the irreduciblecharacters of $G$ and the interpolation points as common zeros of certainfinite subsets of these characters. The consistent use of Lie theoreticalmethods reveals the central ideas clearly and allows for a simple uniformdevelopment of the subject. Furthermore it points to genuine and perhaps farreaching Lie theoretical connections.

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