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

Abstract

We consider the double affine Hecke algebra H=H(k₀,k₁,k₀v,k₁v;q) associated with the root system (C₁v,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 (C₁v,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 → y → z → 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.