Multivariate Chebyshev polynomials

We study multivariate Chebyshev polynomials associated with root systems. Using properties of specialized singular elements Phi(;alpha)(g) corresponding to a root system Delta(g), we construct explicitly the measure weight function gamma. The latter ensures that these polynomials are orthonormal; it defines the scalar product in the function space L-2 (F, gamma) where multivariate U-type Chebyshev polynomials U-g(mu) constitute a basis. The obtained results are illustrated by constructing and studying 2-variate polynomials U-g(mu) for root systems Delta(A2), Delta(B2) and Delta(G2).