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Double Affine Hecke Algebras of Rank 1 and the $Z_3$-Symmetric Askey-Wilson Relations

We consider the double affine Hecke algebra $H=H(k_0,k_1,k^\vee_0,k^\vee_1;q)$ associated with the root system $(C^\vee_1,C_1)$. 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 $\hat H$ that is more general than $H$, called the universal double affine Hecke algebra of type $(C_1^\vee,C_1)$. An advantage of $\hat H$ over $H$ is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism ${\hat H} \to H$. We define some elements $x$, $y$, $z$ in $\hat H$ that get mapped to their counterparts in $H$ by this homomorphism. We give an action of Artin's braid group $B_3$ on $\hat H$ that acts nicely on the elements $x$, $y$, $z$; one generator sends $x\mapsto y\mapsto z \mapsto x$ and another generator interchanges $x$, $y$. Using the $B_3$ action we show that the elements $x$, $y$, $z$ in $\hat H$ satisfy three equations that resemble the $Z_3$-symmetric Askey-Wilson relations. Applying the homomorphism ${\hat H}\to H$ we find that the elements $x$, $y$, $z$ in $H$ satisfy similar relations.