ELSEVIER Nuclear Physics B 558 [PM] (1999) 589-603 www.elsevier.nl/loeate/npe An algebraic approach to the non-symmetric Macdonald polynomial Akinori Nishino 1, Hideaki Ujino 2, Miki Wadati Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan Received 7 April 1999; accepted 6 July 1999 Abstract In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are simultaneous eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them. @ 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 02.10.Nj; 02.30.Gp; 03.65.Fd Keywords: Non-symmetric Macdonald polynomial; Dunld-Cherednik operator; Affine Hecke algebra; Rodrigues formula; Cherednik's scalar product 1. Introduction Since non-trivial quantum many-body effects appear in various fields of modern physics, exactly solvable models in quantum mechanics, i.e. quantum integrable systems, have sometimes played an important role. Among them, the systems with inverse-square long-range interactions which are called the Calogero-Sutherland type models (CS models) [ 1-3] have attracted many physicists and mathematicians. The integrability of the quantum systems is guaranteed through the existence of the same number of independent and mutually commuting conserved operators as their degrees of freedom. The Dunkl-Cherednik operator formulation [4-7] systematically provides the conserved operators of the CS models. From the mathematical point of l E-mail; nishino@monet.phys.s.u-tokyo.ac.jp 2 E-mail: ujino@monet.phys.s.u-tokyo.ac.jp 0550-3213/99/$ - see frontmatter ~) 1999 Published by Elsevier Science B.V. All rights reserved. PII S0550-3213 (99)00407-1