Nonsymmetric Macdonald Polynomials and Demazure Characters

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated finite Lie algebra. Introduction Generalizing the characters of compact simple Lie groups, I. Macdonald associated to each irreducible root system a family of orthogonal polynomials Pλ(q, t) indexed by antidominant weights and invariant under the action of the Weyl group. These polynomials depend rationally on parameters q and t = (ts, tl) and, for particular values of these parameters, reduce to familiar objects in representation theory. (1) When q = ts = tl , they are equal to χλ, the Weyl characters of the corresponding root system. (In particular, they are independent of q .) (2) When q = 0, they are the polynomials that give the values of zonal spherical functions on a semisimple p-adic Lie group relative to a maximal compact subgroup. (3) When ts = qks , tl = qkl , and q tends to 1, they are the polynomials that give the values of zonal spherical functions on a real symmetric space G/K which arise from finite-dimensional spherical representations of G. Here ks , kl are the multiplicities of the short, respectively, long, restricted roots. The nonsymmetric Macdonald polynomials Eλ(q, t) (indexed this time by the entire weight lattice) were first introduced by E. Opdam [O] in the differential setting and then by I. Cherednik [C2] in full generality. Unlike the symmetric polynomials, their representation-theoretical meaning is still unexplored. At present time their main importance consists of the fact that they form the common spectrum of a family of commuting operators (the Cherednik operators) which play a preponderant role in the DUKE MATHEMATICAL JOURNAL Vol. 116, No. 2, c © 2003 Received 8 May 2001. Revision received 19 November 2001. 2000 Mathematics Subject Classification. Primary 33D52; Secondary 17B10, 17B67, 20C08, 33D80.