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Determinantal construction of orthogonal polynomials associated with root systems. (English) Zbl 1048.05081

Introduction: The main objective of this work concerns the explicit computation of families of orthogonal polynomials associated with root systems. Key examples of the families under consideration are the Macdonald polynomials and the Heckman-Opdam generalized Jacobi polynomials. The origin of the Heckman-Opdam polynomials lies in the harmonic analysis of simple Lie groups, where they appear (for special parameter values) as zonal spherical functions on compact symmetric spaces. Other important applications of these polynomials arise in mathematical physics, where they are used to express the eigenfunctions of the quantum Calogero-Sutherland one-dimensional many-body systems. The Macdonald polynomials have similar applications: they appear as zonal spherical functions on compact quantum symmetric spaces, and they are used to express the eigenfunctions of Ruijsenaars’ \((q\)-)difference Calogero-Sutherland systems. Depending on the specific application of interest, our work may thus be viewed as providing an explicit construction for the zonal spherical functions on compact (quantum) symmetric spaces or for the eigenfunctions of the (difference) Calogero-Sutherland type quantum many-body models.
The usual definition of the Heckman-Opdam and Macdonald polynomials involves a Gram-Schmidt type orthogonalization of the monomial basis with respect to a generalized Haar measure. This definition, although most appropriate from a theoretical point of view, is not very adequate for the explicit computation of the polynomials in question. The main result of this paper is a determinantal formula for the Heckman-Opdam and Macdonald polynomials that gives rise to an efficient recursive procedure from which their expansion in the monomial basis can be constructed explicitly. For the type \(A\) root systems the Heckman-Opdam polynomials reduce (in essence) to Jack’s polynomials and the Macdonald polynomials reduce to Macdonald’s symmetric functions. In this case the determinantal construction of the polynomials under consideration was laid out in previous work by L. Lapointe, A. Lascoux and J. Morse [Electron. J. Comb. 7, No. 1, Notes N1 (2000; Zbl 0934.05123) and Int. Math. Res. Not. 1998, No. 18, 957–978 (1998; Zbl 0916.05077)]. More specifically, the results of the present paper constitute a generalization of the methods of Lapointe, Lascoux and Morse [loc. cit.] to the case of arbitrary root systems. For the Heckman-Opdam families we consider general (not necessarily reduced) root systems and general values of the root multiplicity parameters. For the Macdonald families, however, we restrict for technical reasons to those (reduced) root systems for which the dual root system \(R^\vee\) has a minuscule weight (thus including the types \(A_N\) \(B_N\), \(C_N\), \(D_N\), \(E_6\) and \(E_7\) while excluding the types \(BC_n\), \(E_8\), \(F_4\) and \(G_2)\).

MSC:

05E35 Orthogonal polynomials (combinatorics) (MSC2000)
33C52 Orthogonal polynomials and functions associated with root systems
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