
SIGMA 4 (2008), 052, 17 pages arXiv:0711.2320
https://doi.org/10.3842/SIGMA.2008.052
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
Tom H. Koornwinder
Kortewegde Vries Institute, University of Amsterdam,
Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Received November 15, 2007, in final form June 03, 2008; Published online June 10, 2008
Abstract
This paper builds on the previous paper by the author,
where a relationship
between Zhedanov's algebra AW(3) and the double affine Hecke algebra
(DAHA)
corresponding to the AskeyWilson polynomials was established.
It is shown here that the spherical subalgebra of this DAHA is
isomorphic to AW(3) with an additional relation that the Casimir operator
equals an explicit constant.
A similar result with qshifted parameters holds for the
antispherical subalgebra.
Some theorems on centralizers and centers for the
algebras under consideration will finally be proved as corollaries of
the characterization of the spherical and antispherical subalgebra.
Key words:
Zhedanov's algebra AW(3); double affine Hecke algebra in rank one; AskeyWilson polynomials; spherical subalgebra.
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