A new scalar product for nonsymmetric Jack polynomials

Jack polynomials are a remarkable family of polynomials in n variables x = (x1, · · · , xn) with coefficients in the field F := Q(α) where α is an indeterminate. They arise naturally in several statistical, physical, combinatorial, and representation theoretic considerations. The symmetric polynomials ([M1], [St], [LV], [KS]) Jλ = J (α) λ are indexed by partitions λ = (λ1, · · · , λn) where λ1 ≥ · · · ≥ λn ≥ 0. The nonsymmetric polynomials Fη = F (α) η ([Op], [KS], and §2) are indexed by compositions η = (η1, · · · , ηn) where ηi ≥ 0 are integers. They constitute orthogonal bases, respectively, for symmetric polynomials and all polynomials, with respect the scalar product 〈f, g〉0 ≡ ∫