Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the non-symmetric ones. These formulas are then implemented by a closed expression of symmetric and non-symmetric Jack polynomials in terms of certain tableaux. The main application is a proof of a conjecture of Macdonald stating certain integrality and positivity properties of Jack polynomials.Comment: Preprint March 1996, to appear in Invent. Math., 15 pages, Plain Te

Abstract. We introduce families of symmetric and non-symmetric polynomials (the quantum Capelli polynomials) which depend on two parameters q and t. They are defined in terms of vanishing conditions. In the differential limit (q = t α and t → 1) they are related to Capelli identities. It is shown that the quantum Capelli polynomials form an eigenbasis for certain q-difference operators. As a corollary, we obtain that the top homogeneous part is a symmetric/non-symmetric Macdonald polynomial. Furthermore, we study the vanishing and integrality properties of the quantum Capelli polynomials.Mathematics Subject Classification (1991). 05E05, 12H10, 39A70.

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