The Relationship between Zhedanov's Algebra $AW(3)$ and the Double Affine Hecke Algebra in the Rank One Case

Tom H. Koornwinder

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

Zhedanov's algebra $AW(3)$ is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order $q$-difference operator for the Askey–Wilson polynomials. It is proved that this representation is faithful for a certain quotient of $AW(3)$ such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey–Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of $AW(3)$ is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.

Détail

Keywords: Zhedanov's algebra $AW(3)$; double affine Hecke algebra in rank one; Askey–Wilson polynomials; non-symmetric Askey–Wilson polynomials.

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