The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case

Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.


Introduction
Zhedanov [16] introduced in 1991 an algebra AW (3) with three generators K 0 , K 1 , K 2 and three relations in the form of q-commutators, which describes deeper symmetries of the Askey-Wilson polynomials. In fact, for suitable choices of the structure constants of the algebra, the Askey-Wilson polynomial p n (x) is the kernel of an intertwining operator between a representation of AW (3) by q-difference operators on the space of polynomials in x and a representation by tridiagonal operators on the space of of infinite sequences (c n ) n=1,2,... . In the first representation K 1 is multiplication by x and K 0 is the second order q-difference operator for which the Askey-Wilson polynomials are eigenfunctions with explicit eigenvalues λ n . In the second representation K 0 is the diagonal operator with diagonal elements λ n and K 1 is the tridiagonal operator corresponding to the three-term recurrence relation for the Askey-Wilson polynomials. The formula for p n (x) expressing the intertwining property with respect to K 2 is the so-called q-structure relation for the Askey-Wilson polynomials (see [6]) and the relation for AW (3) involving the q-commutator of K 1 and K 2 is the so-called q-string equation (see [4]). Terwilliger & Vidunas [15] showed that every Leonard pair satisfies the AW (3) relations for a suitable choice of the structure constants.
In 1992, one year after Zhedanov's paper [16], Cherednik [2] introduced double affine Hecke algebras associated with root systems (DAHA's). This was the first of an important series of papers by the same author, where a representation of the DAHA was given in terms of qdifference-reflection operators (q-analogues of Dunkl operators), joint eigenfunctions of such operators were identified as non-symmetric Macdonald polynomials, and Macdonald's conjectures for ordinary (symmetric) Macdonald polynomials associated with root systems could be proved. For a nice exposition of this theory see Macdonald's recent book [7]. In particular, the DAHA approach to Macdonald-Koornwinder polynomials, due to several authors (see Sahi [11,12], Stokman [14] and references given there) is also presented in [7]. The last chapter of [7] discusses the rank one specialization of these general results. For the DAHA of type A 1 (one parameter) this yields non-symmetric q-ultraspherical polynomials. For the DAHA of type (C ∨ 1 , C 1 ) (four parameters) the non-symmetric Askey-Wilson polynomials are obtained. These were earlier treated by Sahi [12] and by Noumi & Stokman [9]. See also Sahi's recent paper [13].
Comparison of Zhedanov's AW (3) with the DAHA of type of type (C ∨ 1 , C 1 ), denoted byH, suggests some relationship. Both algebras are presented by generators and relations, the first has a representation by q-difference operators on the space of symmetric Laurent polynomials in z and the second has a representation by q-difference-reflection operators on the space of general Laurent polynomials in z. Since this representation of the DAHA is called the basic representation ofH, I will call the just mentioned representation of AW (3) also the basic representation. In the basic representation of AW (3) the operator K 0 is equal to some operator D occurring in the basic representation ofH and involving reflections, provided D is restricted in its action to symmetric Laurent polynomials. This suggests that the basic representation of AW (3) may remain valid if we represent K 0 by D, so that it involves reflection terms. It will turn out in this paper that this conjecture is correct in the A 1 case, i.e., when the Askey-Wilson parameters are restricted to the continuous q-ultraspherical case. In the general case the conjecture is true for a rather harmless central extension of AW (3) involving a generator T 1 , which will be identified with the familiar T 1 inH which has in the basic representation ofH the symmetric Laurent polynomials as one of its two eigenspaces. This paper does not suppose any knowledge about the general theory of double affine Hecke algebras and about Macdonald and related polynomials in higher rank. The contents of the paper are as follows. Section 2 presents AW (3) and its relationship with Askey-Wilson polynomials. We add to AW (3) one more relation expressing that the Casimir operator Q is equal to a special constant Q 0 (of course precisely the constant occurring for Q in the basic representation), and we denote the resulting quotient algebra by AW (3, Q 0 ). Then it is shown that the basic representation of AW (3, Q 0 ) is faithful. Section 3 discussesH (the DAHA of type (C ∨ 1 , C 1 )), its basic representation, and the basis vectors for the 2-dimensional eigenspaces of the operator D in terms of Askey-Wilson polynomials. Section 4 gives an explicit expression for the non-symmetric Askey-Wilson polynomials which is in somewhat different terms than the explicit expression in [7, § 6.6]. Two presentations ofH by generators and relations of PBW-type are given in Section 5. The faithfulness of the basic representation is proved (a result which of course is also a special case of the known result in the case of general rank, see Sahi [11]). The main result of the present paper, the embedding of a central extension of AW (3, Q 0 ) inH, is stated and proved in Section 6.
For the computations in this paper I made heavy use of computer algebra performed in Mathematica R . For reductions of expressions in non-commuting variables subject to relations I used the package NCAlgebra [8] within Mathematica R . Mathematica notebooks containing these computations will be available for downloading in http://www.science.uva.nl/~thk/art/. For (q-)Pochhammer symbols and (q-)hypergeometric series use the notation of [3]. In particular,
For Laurent polynomials f in z the z-dependence will be written as f [z]. Symmetric Laurent

Zhedanov's algebra AW (3)
Zhedanov [16] introduced an algebra AW (3) with three generators K 0 , K 1 , K 2 and with three relations is the q-commutator and where the structure constants B, C 0 , C 1 , D 0 , D 1 are fixed complex constants. He also gave a Casimir operator which commutes with the generators. Clearly, AW (3) can equivalently be described as an algebra with two generators K 0 , K 1 and with two relations Then the Casimir operator Q can be written as Let the structure constants be expressed in terms of a, b, c, d by means of e 1 , e 2 , e 3 , e 4 (see (1.2)) as follows:

4)
Then there is a representation (the basic representation) of the algebra AW (3) with structure constants (2.4) on the space A sym of symmetric Laurent polynomials f [z] = f [z −1 ] as follows: where D sym , given by is the second order operator having the Askey-Wilson polynomials (see [1], as eigenfunctions. It can indeed be verified that the operators K 0 , K 1 given by (2.5) satisfy relations (2.1), (2.2) with structure constants (2.4), and that the Casimir operator Q becomes the following constant in this representation: where − q 2 (e 2 e 4 + 2e 4 + e 2 ) + q(e 2 3 − 2e 2 e 4 − e 1 e 3 ) + e 4 (1 − e 2 ) . (2.8) Let AW (3, Q 0 ) be the algebra generated by K 0 , K 1 with relations (2.1), (2.2) and assuming the structure constants (2.4). Then the basic representation of AW (3) is also a representation of AW (3, Q 0 ). The Askey-Wilson polynomials are given by (ab, ac, ad; q) n a n 4 φ 3 q −n , q n−1 abcd, az, az −1 ab, ac, ad ; q, q . (2.10) These polynomials are symmetric in a, b, c, d (although this cannot be read off from (2.10)). We will work with the renormalized version which is monic as a Laurent polynomial in z (i.e., the coefficient of z n equals 1): Note that the monic Askey-Wilson polynomials P n [z] are well-defined for all n under condition (1.1). The eigenvalue equation involving D sym is D sym P n = λ n P n , λ n := q −n + abcdq n−1 . (2.14) .
Proof . Because of Lemma 2.1 it is sufficient to show that the operators acting on A sym are linearly independent. By (2.12) and (2.13) we have for all j: for certain coefficients a k,l , b k,l such that for some l a m,l = 0 or b m,l = 0. Then it follows from (2.18) that for all j, when we let the left-hand side of (2.19) act on By (2.12) we have, writing x = q j+m and u = q −1 abcd, We can consider the identity (2.20) as an identity for Laurent polynomials in x. Since the left-hand side vanishes for infinitely many values of x, it must be identically zero. Let n be the maximal l for which a m,l = 0 or b m,l = 0. Then, in particular, the coefficients of x −n and x n in the left-hand side of (2.20) must be zero. This gives explicitly: a m,n + qb m,n = 0, u n a m,n + q −1 u n b m,n = 0.
This implies a m,n = b m,n = 0, contradicting our assumption.
Conditions on q, a, b, c, d in [7] are more strict than in (1.1). This will give no problem, as can be seen by checking all results hereafter from scratch. From (3.1) and (3.2) and the non-vanishing of a, b, c, d we see that T 1 and T 0 are invertible: ( The representation property is from [7, § 6.4] or by straightforward computation. The faithfulness is from [7, (4.7.4)] or by an independent proof later in this paper. Now we can compute: If we compare (3.14) and (2.6) then we see that In particular, if we apply D to the Askey-Wilson polynomial P n [z] given by (2.11) then we obtain from (2.12) that DP n = λ n P n .
for some symmetric Laurent polynomial g.
Proof . We compute which settles the first assertion. We also compute This for some Laurent polynomial g and we obtain g Proposition 3.2. T 0 given by (3.12) has eigenvalues −q −1 cd and −1.
for some Laurent polynomial g satisfying g Proof . We compute which settles the first assertion. We also compute Then the second assertion is proved by similar arguments as in the proof of Proposition 3.1.
We now look for further explicit solutions of the eigenvalue equation Clearly, the solution P n (see (3.15)) also satisfies T 1 P n = −ab P n . In order to find further solutions of (3.16) we make an Ansatz for f as suggested by Propositions 3.1 and 3.2, namely , in each case with g symmetric. Then it turns out that (3.16) takes the form of the Askey-Wilson second order q-difference equation, but with parameters and sometimes also the degree changed. We thus obtain as further solutions f of (3.16) for n ≥ 1: So we have for n ≥ 1 four different eigenfunctions of D at eigenvalue q −n + abcdq n−1 which are also eigenfunction of T 1 or T 0 : They all are Laurent polynomials of degree n with highest term z n and lowest term const z −n : Since the eigenvalues λ n are distinct for different n, it follows that D has a 1-dimensional eigenspace A 0 at eigenvalue λ 0 , consisting of the constant Laurent polynomials, and that it has a 2-dimensional eigenspace A n at eigenvalue λ n if n ≥ 1, which has P n and P † n as basis vectors, but which also has any other two out of P n , Q n , P † n , Q † n as basis vectors, provided these two functions have the coefficients of z −n distinct. Generically we can use any two out of these four as basis vectors. The basis consisting of P n and P † n occurs in [7, § 6.6]. In the following sections we will work first with the basis consisting of P n and Q † n , but afterwards it will be more convenient to use P n and Q n .

Non-symmetric Askey-Wilson polynomials
Since T 1 and T 0 commute with D, the eigenspaces of D in A are invariant under Y = T 1 T 0 . We can find explicitly the eigenvectors of Y within these eigenspaces A n .
span the one-dimensional eigenspaces of Y within A n with the following eigenvalues:
Remark 4.2. By condition (1.1) all eigenvalues of Y on A (see (4.4), (4.5)) are distinct. So for all n ∈ Z E n [z] is the unique Laurent polynomial of degree |n| which satisfies (4.4) or (4.5) and has coefficient of z n equal to 1. Moreover, for n ≥ 1, E −n is the unique element of A n of the form (4.6), and E n is the unique element of A n of the form (4.7) Next, (4.9), (4.10) and (3.20) yield (4.12)

A PBW-type theorem forH
In this section I will give two other sets of relations forH, both equivalent to (3.1)-(3.4) and both of PBW-type form. For the second set of relations we will see that the spanning set of elements ofH, as implied by these relations, is indeed a basis. This is done by showing that this set of elements is linearly independent in the basic representation, which also shows that this representation is faithful. The faithfulness of the basic representation was first shown, in the more general n variable setting, by Sahi [11].
H is spanned by the elements Z m Y n T i 1 , where m, n ∈ Z, i = 0, 1.
The last statement follows from the PBW-type structure of the relations (5.7). Observe that by the first five relations together with the trivial relations, every word in T 1 , Y , Y −1 , Z, Z −1 can be written as a linear combination of words with at most one occurrence of T 1 in each word and only on the right, and with no substrings Y Y −1 , Y −1 Y , ZZ −1 , Z −1 Z, and with no more occurrences of Y , Y −1 , Z, Z −1 in each word than in the original word. If in one of these terms there are misplacements (Y or Y −1 before Z or Z −1 ) then apply one of the last four relations followed by the previous step in order to reduce the number of misplacements. Proof . For j > 0 we have This follows from (4.4)-(4.7), (4.11) and (3.5). Suppose that some linear combination acts as the zero operator in the basic representation, while not all coefficients a m,n , b m,n are zero. Then there is a maximal r for which a r,n or b r,n is nonzero for some n. If b r,n = 0 for some n then let the operator (5.9) act on E −j . By (5.8) we have that for all j ≥ 1 n b r,n (q j−1 abcd) n z r+j = 0, hence n b r,n (q j−1 abcd) n = 0.
By assumption (1.1) we see that n b r,n w n = 0. Hence b r,n = 0 for all n, which is a contradiction. So a r,n = 0 for some n. Let the operator (5.9) act on T −1 1 E −j . By (5.8) we have that for all j ≥ 1 n a r,n (q j−1 abcd) n z r+j = 0, hence n a r,n (q j−1 abcd) n = 0.
Again we arrive at the contradiction that a r,n = 0 for all n.
6 The embedding of a central extension of AW (3, Q 0 ) inH Let us now examine whether the representation (2.5) of AW (3) on A sym extends to a representation on A if we let K 0 act as D instead of D sym . It will turn out that this is only true for certain specializations of a, b, c, d, but that a suitable central extension AW (3) of AW (3) involving T 1 will realize what we desire.
Theorem 6.2. There is a representation of the algebra AW (3, Q 0 ) on the space A of Laurent polynomials f [z] such that K 0 acts as D, K 1 acts by multiplication by z + z −1 , and the action of T 1 is given by (3.11). This representation is faithful.
Proof . It follows by straightforward computation, possibly using computer algebra, that this is a representation of AW (3, Q 0 ). In the same way as for Lemma 2.1 it can be shown that AW (3, Q 0 ) is spanned by the elements (m, n = 0, 1, 2, . . . , i, j = 0, 1). (6.8) Now we will prove that the representation is faithful. Suppose that for certain coefficients a k,l , b k,l , c k,l , d k,l we have