SIGMA 6 (2010), 065, 9 pages      arXiv:1001.2764      https://doi.org/10.3842/SIGMA.2010.065

Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations

Tatsuro Ito ^a and Paul Terwilliger ^b
a) Division of Mathematical and Physical Sciences, Graduate School of Natural Science nd Technology, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
^b) Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706-1388, USA

Received January 23, 2010, in final form August 10, 2010; Published online August 17, 2010

Abstract
We consider the double affine Hecke algebra H=H(k₀,k₁,k₀^v,k₁^v;q) associated with the root system (C₁^v,C₁). We display three elements x, y, z in H that satisfy essentially the Z₃-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra H^{^} that is more general than H, called the universal double affine Hecke algebra of type (C₁^v,C₁). An advantage of H^{^} over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism H^{^} → H. We define some elements x, y, z in H^{^} that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B₃ on H^{^} that acts nicely on the elements x, y, z; one generator sends x → y → z → x and another generator interchanges x, y. Using the B₃ action we show that the elements x, y, z in H^{^} satisfy three equations that resemble the Z₃-symmetric Askey-Wilson relations. Applying the homomorphism H^{^} → H we find that the elements x, y, z in H satisfy similar relations.

Key words: Askey-Wilson polynomials; Askey-Wilson relations; braid group.

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