Multivariable Wilson polynomials and degenerate Hecke algebras

Abstract

Abstract. We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type (C ∨ n , Cn). A certain representation in terms of difference-reflection operators naturally leads to the definition of nonsymmetric versions of the multivariable Wilson polynomials. Using the degenerate Hecke algebra we derive several properties, such as orthogonality relations and quadratic norms, for the nonsymmetric and symmetric multivariable Wilson polynomials. Mathematics Subject Classification (2000). Primary 33C52, 33C80; Secondary 20C08.

The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

“…Remark 4. While A 3 coincides with a degenerate double affine Hecke algebra of type ( Č1 , C 1 ) [13], it appears that A n does not coincide with the higher rank version of this degenerate double affine Hecke algebra of type ( Čn−2 , C n−2 ), which has been investigated in [15,16].…”

The non-symmetric Wilson polynomials are the Bannai–Ito polynomials

“…First, given the applications of the one-variable Bannai-Ito polynomials to exactly solvable models (e.g. [6,7]), it would be of interest to study the multivariate non-symmetric Wilson polynomials introduced in [19] from that perspective; the results of such an investigation should be compared for example with those found in [27] and [28]. Second, in view of the fact that the Bannai-Ito polynomials B n (x) obeying a discrete and finite orthogonality relation arise as Racah coefficients for positive-discrete series representations of osp(1|2) [8], it would be natural to look for a similar interpretation for the modified Bannai-Ito polynomials Q n (x), which satisfy a continuous orthogonality relation.…”

“…These results relate to the large body of work on multivariate orthogonal polynomials and generalized Calogero-Moser systems associated with Cherednik algebras and root systems; see for example [14,15,16,17]. In the present paper, we shall put the Bannai-Ito algebra and polynomials in a similar framework by exhibiting their relationship with the degenerate double affine Hecke algebra associated to the rank one root system of type (C ∨ 1 , C 1 ) and with the non-symmetric Wilson polynomials introduced by Groenevelt in [18,19].…”

Discrete harmonic analysis on a Weyl alcove

“…(This algebra is not to be confused with the trigonometric degenerate double affine Hecke algebra represented by the Dunkl operators for the trigonometric Calogero-Sutherland system diagonalized by the Heckman-Opdam multivariate Jacobi polynomials [47,46], or with the q → 1 degenerate double affine Hecke algebra represented by the Dunkl-Cherednik type operators for the confined rational Ruijsenaars system with hyperoctahedral symmetry diagonalized by the multivariate Wilson polynomials [24,8,9].) The Bethe Ansatz method is used to show that our Laplacians are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions.…”