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.


Introduction
The double affine Hecke algebra (DAHA) for a reduced root system was defined by Cherednik [2], and the definition was extended to include nonreduced root systems by Sahi [11]. The most general DAHA of rank 1 is associated with the root system (C ∨ 1 , C 1 ) [8]; this algebra involves five nonzero parameters and will be denoted by H = H(k 0 , k 1 , k ∨ 0 , k ∨ 1 ; q). We mention some recent results on H. In [12] Sahi links certain H-modules to the Askey-Wilson polynomials [1]. This link is given a comprehensive treatment by Noumi and Stokman [7]. In [9] Oblomkov and Stoica describe the finite-dimensional irreducible H-modules under the assumption that q is not a root of unity. In [8] Oblomkov gives a detailed study of the algebraic structure of H, and finds an intimate connection to the geometry of affine cubic surfaces. His point of departure is the case q = 1; under that assumption he finds that the spherical subalgebra of H is generated by three elements X 1 , X 2 , X 3 that mutually commute and satisfy a certain cubic equation [8, Theorem 2.1, Proposition 3.1]. In [4,5] Koornwinder describes the spherical subalgebra of H under the assumption that q is not a root of unity. His main results [4,Corollary 6.3], [5,Theorem 3.2] are similar in nature to those of Oblomkov, although he formulates these results in a very different way and works with a different presentation of H. In Koornwinder's formulation the spherical subalgebra of H is related to the Askey-Wilson algebra AW (3), which was introduced by Zhedanov in [17]. The original presentation of AW (3) involves three generators and three relations [17, lines (1.1a)-(1.1c)]. Koornwinder works with a slightly different presentation for AW (3) that involves two generators and two relations [4, lines (2.1), (2.2)]. These two relations are sometimes called the Askey-Wilson relations [15]. For the algebra AW (3) a third presentation is known [10, p. 101], [14], [16,Section 4.3] and described as follows. For a sequence of scalars g x , g y , g z , h x , h y , h z the corresponding Askey-Wilson algebra is defined by generators X, Y , Z and relations We will refer to (1)-(3) as the Z 3 -symmetric Askey-Wilson relations. Upon eliminating Z in (2), (3) using (1) we obtain the Askey-Wilson relations in the variables X, Y . Upon sub- we recover the the original presentation for AW (3) in the variables X, Y , Z ′ . In this paper we return to the elements X 1 , X 2 , X 3 considered by Oblomkov, although for notational convenience we will call them x, y, z. We show that x, y, z satisfy three equations that resemble the Z 3 -symmetric Askey-Wilson relations. The resemblance is described as follows. The equations have the form (1)-(3) with h x , h y , h z not scalars but instead rational expressions involving an element t 1 that commutes with each of x, y, z. The element t 1 appears earlier in the work of Koornwinder [4, Definition 6.1]; we will say more about this at the end of Section 2. Our derivation of the three equations is elementary and illuminates a role played by Artin's braid group B 3 .
Our proof is summarized as follows. Adapting some ideas of Ion and Sahi [3] we work with an algebraĤ that is more general than H, called the universal double affine Hecke algebra (UDAHA) of type (C ∨ 1 , C 1 ). An advantage ofĤ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphismĤ → H. We define some elements x, y, z inĤ that get mapped to their counterparts in H by this homomorphism. Adapting [3, Theorem 2.6] we give an action of the braid group B 3 onĤ 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 inĤ satisfy three equations that resemble the Z 3 -symmetric Askey-Wilson relations. Applying the homomorphismĤ → H we find that the elements x, y, z in H satisfy similar relations.
2 The double af f ine Hecke algebra of type (C ∨ 1 , C 1 ) Throughout the paper F denotes a field. An algebra is meant to be associative and have a 1. We recall the double affine Hecke algebra of type (C ∨ 1 , C 1 ). For this algebra there are several presentations in the literature; one involves three generators [4,5,13] and another involves four generators [6, p. 160], [7,8,9]. We will use essentially the presentation of [6, p. 160], with an adjustment designed to make explicit the underlying symmetry.
This algebra is called the double affine Hecke algebra (or DAHA) of type (C ∨ 1 , C 1 ).
We now state our main result. In this result part (ii) follows from [8, Theorem 2.1]; it is included here for the sake of completeness.
Then the following (i)-(iv) hold: (i) t 1 commutes with each of x, y, z.
(iii) Assume q 2 = 1 and q 4 = 1. Then Char(F) = 2 and Then The equations in Theorem 2.4(iv) resemble the Z 3 -symmetric Askey-Wilson relations, as we discussed in Section 1.
We will prove Theorem 2.4 in Section 5. We comment on how Theorem 2.4 is related to the work of Koornwinder [4]. Define x, y, z as in Theorem 2.4. Then that theorem describes how x, y, z, t 1 are related. If we translate [4, Definition 6.1, Corollary 6.3] into the presentation of Definition 2.1, then it describes how x, y, t 1 are related, assuming q is not a root of unity and some constraints on k 0 , k 1 , k ∨ 0 , k ∨ 1 . Under these assumptions and modulo the translation the following coincide: (i) the main relations [4, lines (6.2), (6.3)] of [4, Definition 6.1]; (ii) the relations obtained from the last two equations of Theorem 2.4(iv) by eliminating z using the first equation. 3 The universal double af f ine Hecke algebra of type (C ∨ 1 , C 1 ) In our proof of Theorem 2.4 we will initially work with a homomorphic preimageĤ of H(k 0 , k 1 , k ∨ 0 , k ∨ 1 ; q) called the universal double affine Hecke algebra of type (C ∨ 1 , C 1 ). Before we get into the details, we would like to acknowledge howĤ is related to the work of Ion and Sahi [3]. Given a general DAHA (not just rank 1) Ion and Sahi construct a groupÃ called the double affine Artin group [3, Definition 3.4, Theorem 3.10]. The given DAHA is a homomorphic image of the group F-algebra FÃ [3, Definition 1.13]. For the case (C ∨ 1 , C 1 ) of the present paper, their homomorphism has a factorization FÃ →Ĥ → H(k 0 , k 1 , k ∨ 0 , k ∨ 1 ; q). In this section and the next we will obtain some facts aboutĤ. We could obtain these facts from [3] by applying the homomorphism FÃ →Ĥ, but for the purpose of clarity we will prove everything from first principles.
We now defineĤ and describe some of its basic properties. In Section 4 we will discuss how the group B 3 acts onĤ. In Section 5 we will use the B 3 action to prove Theorem 2.4. Definition 3.1. LetĤ denote the F-algebra defined by generators t ±1 i , (t ∨ i ) ±1 (i = 0, 1) and relations We callĤ the universal double affine Hecke algebra (or UDAHA) of type (C ∨ 1 , C 1 ).
Definition 3.3. Observe that inĤ the element t ∨ 0 t 0 t ∨ 1 t 1 is invertible; let Q denote the inverse.
One advantage ofĤ over H(k 0 , k 1 , k ∨ 0 , k ∨ 1 ; q) is thatĤ has more automorphisms. This is illustrated in the next lemma. By an automorphism ofĤ we mean an F-algebra isomorphism H →Ĥ.
Lemma 3.5. There exists an automorphism ofĤ that sends This automorphism fixes Q.
Proof . The result follows from Definition 3.1, once we verify that t 0 t ∨ 1 t 1 t ∨ 0 = Q −1 . This equation holds since each side is equal to t ∨−1 Lemma 3.6. In the algebraĤ the element Q −1 is equal to each of the following: Proof . To each side of the equation t ∨ 0 t 0 t ∨ 1 t 1 = Q −1 apply three times the automorphism from Lemma 3.5. Definition 3.7. We define elements x, y, z inĤ as follows.
The following result suggests why x, y, z are of interest.
Lemma 3.8. Let u, v denote invertible elements in any algebra such that each of u + u −1 , v + v −1 is central. Then (ii) uv + (uv) −1 commutes with each of u, v.
Proof . (i) Observe that In these equations the expressions on the right are equal since u + u −1 and v + v −1 are central. The result follows.
Corollary 3.9. In the algebraĤ the element t 1 commutes with each of x, y, z.

The braid group B 3
In this section we display an action of the braid group B 3 on the algebraĤ from Definition 3.1. This B 3 action will be used to prove Theorem 2.4.
Proof . There exists an automorphism A ofĤ that sends h → t −1 1 ht 1 for all h ∈Ĥ. Define Note that T ∨ 0 , T 0 , T ∨ 1 , T 1 are invertible and that In each of these four equations the expression on the right is central so the expression on the left is central. Using (11) and Lemma 3.6, By these comments there exists an F-algebra homomorphism B :Ĥ → H that sends We claim that B 3 = A. To prove the claim we show that B 3 , A agree at each of t ∨ 0 , t 0 , t ∨ 1 , t 1 . Note that A fixes t 1 . Note also that t 1 is fixed by B and hence B 3 ; therefore B 3 and A agree at t 1 . The map B sends We have shown B 3 = A. By this and since A is invertible, we see that B is invertible and hence an automorphism ofĤ. Define Note that S ∨ 0 , S 0 , S ∨ 1 , S 1 are invertible and In each of these four equations the expression on the right is central so the expression on the left is central. Using (12) and Lemma 3.6, By these comments there exists an F-algebra homomorphism C :Ĥ→Ĥ that sends We claim that C 2 = A. To prove the claim we show that C 2 , A agree at each of t ∨ 0 , t 0 , t ∨ 1 , t 1 .
In the above line the expression on the right equals t −1 1 t 0 t 1 . To see this, note that t ∨ 1 t 1 t ∨ 0 t 0 = t 0 t ∨ 1 t 1 t ∨ 0 since each side equals Q −1 by Lemma 3.6. We have shown that C 2 , A agree at t 0 . By the above comments C 2 , A agree at each of t ∨ 0 , t 0 , t ∨ 1 , t 1 so C 2 = A. Therefore C is invertible and hence an automorphism ofĤ. We have shown that the desired B 3 action exists.
The next result is immediate from Lemma 4.2 and its proof.
The proof of Theorem 2.4 Recall the elements x, y, z ofĤ from Definition 3.7. In this section we describe how the group B 3 acts on these elements. Using this information we show that x, y, z satisfy three equations that resemble the Z 3 -symmetric Askey-Wilson relations. Using these equations we obtain Theorem 2.4.
Theorem 5.1. The B 3 action from Lemma 4.2 does the following to the elements x, y, z from Definition 3.7. The generator a fixes each of x, y, z. The generator b sends x → y → z → x.
The generator c swaps x, y and sends z → z ′ where Proof . The generator a fixes each of x, y, z by Corollary 3.9 and since a(h) = t −1 1 ht 1 for all h ∈Ĥ. The generator b sends x → y → z → x by Definition 3.7, Corollary 3.9, and Lemma 4.2. Similarly the generator c swaps x, y. Define z ′ = c(z). We show that z ′ satisfies the equations in the theorem statement. We first show that . By this and Definition 3.3, By Lemma 3.6, Using (14), (15) and y = t ∨ 1 t 1 + (t ∨ 1 t 1 ) −1 we obtain (13). Next we show that By Lemma 4.3, Combining this with (13) we obtain (16) after a brief calculation. In (13) we multiply each term on the right by t 1 and use c(t 1 ) = t 1 to get In (16) we multiply each term on the left by t −1 1 and use c(t −1 1 ) = t −1 1 together with the fact that y commutes with t 1 to get We have since both sides equal t −1 1 (t ∨ 1 + t ∨−1 1 )t 1 . We now add (17), (18) and simplify the result using (19) to obtain We now apply c to each side of (20) and evaluate the result. To aid in this evaluation we recall that c swaps x, y; also c swaps z, z ′ since c 2 = a and a(z) = z. By these comments and Lemma 4.3 we obtain

8
T. Ito and P. Terwilliger Theorem 5.2. In the algebraĤ the elements x, y, z are related as follows: Proof . To get the first equation, eliminate z ′ from the equations of Theorem 5.1. To get the other two equations use the B 3 action from Lemma 4.2. Specifically, apply b twice to the first equation and use the data in Lemma 4.3, together with the fact that b cyclically permutes x, y, z.