The binomial formula for nonsymmetric Macdonald polynomials

The q-binomial theorem is essentially the expansion of (x − 1)(x − q) · · · (x − q) in terms of the monomials x. In a recent paper [O], A. Okounkov has proved a beautiful multivariate generalization of this in the context of symmetric Macdonald polynomials [M1]. These polynomials have nonsymmetric counterparts [M2] which are of substantial interest, and in this paper we establish nonsymmetric analogues of Okounkov’s results. An integral vector v ∈ Z is called “dominant” if v1 ≥ · · · ≥ vn; and it is called a “composition” if vi ≥ 0, for all i. To avoid ambiguity we reserve the letters u, v for integral vectors, α, β, γ for compositions, and λ, μ for “partitions” (dominant compositions).