Double Affine Hecke Algebras of Rank 1 and the Z ₃ -Symmetric Askey-Wilson Relations

We consider the double affine Hecke algebra H = H(k 0 , k 1 , k 0 , k 1 ; q) associated with the root system (C 1 , C 1 ). We display three elements x, y, z in H that satisfy essentially the Z 3 -symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra that is more general than H, called the universal double affine Hecke algebra of type (C 1 , C 1 ). An advantage of over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism H. We define some elements x, y, z in that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B 3 on that acts nicely on the elements x, y, z; one generator sends x y z x and another generator interchanges x, y. Using the B 3 action we show that the elements x, y, z in satisfy three equations that resemble the Z 3 -symmetric Askey-Wilson relations. Applying the homomorphism H we find that the elements x, y, z in H satisfy similar relations.