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 n -dimensional Euclidean space, we construct a sequence of polynomials P k : R n → R n {P_k}:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n} so that the mapping P k {P_k} is conjugate to stretching the figure by a factor k k and folding it back onto itself. If n = 1 n = 1 and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials P k {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.