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

[摘要] We consider the double affine Hecke algebra H = H ( k 0 , k 1 , k 0 v , k 1 v ; q ) associated with the root system ( C 1 v , 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 H ^ that is more general than H , called the universal double affine Hecke algebra of type ( C 1 v , C 1 ). 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 B 3 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 B 3 action we show that the elements x , y , z in H ^ satisfy three equations that resemble the Z 3 -symmetric Askey-Wilson relations. Applying the homomorphism H ^ → H we find that the elements x , y , z in H satisfy similar relations.

[关键词] Askey–Wilson polynomials;Askey–Wilson relations;braid group