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

We consider the double affine Hecke algebra H = H(k0, k1, k∨ 0 , k ∨ 1 ; q) associated with the root system (C∨ 1 , 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 Ĥ that is more general than H, called the universal double affine Hecke algebra of type (C∨ 1 , C1). An advantage of Ĥ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism Ĥ → H. We define some elements x, y, z in Ĥ that get mapped to their counterparts in H by this homomorphism. We give an action of Artin’s braid group B3 on Ĥ that acts nicely on the elements x, y, z; one generator sends x 7→ y 7→ z 7→ x and another generator interchanges x, y. Using the B3 action we show that the elements x, y, z in Ĥ satisfy three equations that resemble the Z3-symmetric Askey–Wilson relations. Applying the homomorphism Ĥ → H we find that the elements x, y, z in H satisfy similar relations.