Generalized Chebyshev Polynomials Associated with Affine Weyl Groups

We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in n-dimensional Euclidean space, we construct a sequence of polynomials P/c: Rn —► Rn so that the mapping P^ is conjugate to stretching the figure by a factor A; and folding it back onto itself. If re = 1 and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials Pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.