Generalized Chebyshev polynomials associated with affine Weyl groups

Abstract:

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 ${P_k}:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n}$ so that the mapping ${P_k}$ is conjugate to stretching the figure by a factor $k$ and folding it back onto itself. If $n = 1$ and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials ${P_k}$ are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.