Multivariate Chebyshev polynomials

We study multivariate Chebyshev polynomials associated with root systems. Using properties of specialized singular elements Φg;a corresponding to a root system Δg, we construct explicitly the measure weight function γ. The latter ensures that these polynomials are orthonormal; it defines the scalar product in the function space L2 (F, γ) where multivariate U-type Chebyshev polynomials Uμg constitute a basis. The obtained results are illustrated by constructing and studying 2-variate polynomials Uμg for root systems ΔA2, ΔB2 and ΔG2. (paper)