Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra

This paper builds on the previous paper arXiv:math/0612730 by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra.


Introduction
Zhedanov [15] 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 the basic representation (or polynomial representation) of AW (3) on the space of symmetric Laurent polynomials in z, K 0 acts as the second order q-difference operator D sym for which the Askey-Wilson polynomials are eigenfunctions and K 1 acts as multiplication by z+z −1 .The Casimir operator Q for AW (3) becomes a scalar Q 0 in this representation.Let AW (3, Q 0 ) be AW (3) with the additional relation Q = Q 0 .Then the basic representation AW (3, Q 0 ) is faithful, see [7].There is a parameter changing anti-algebra isomorphism of AW (3) which interchanges K 0 and K 1 , and hence interchanges D sym and z+z −1 in the basic representation.In the basic representation this duality isomorphism can be realized by an integral transform having the Askey-Wilson polynomial P n [z] as kernel which maps symmetric Laurent polynomials to infinite sequences {c n } n=0,1,... .In 1992 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 q-difference-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.The idea of nonsymmetric polynomials was very fruitful, on the one hand as an important extension of traditional harmonic analysis involving orthogonal systems of special functions, on the other hand because the nonsymmetric point of view was helpful for understanding the symmetric case better.For instance, a duality anti-algebra isomorphism and shift operators (as studied earlier by Opdam [11] in the symmetric framework) occur naturally in the non-symmetric context.
Related to Askey-Wilson polynomials the DAHA of type (C ∨ 1 , C 1 ) (four parameters) was studied by Sahi [13,14], Noumi & Stokman [9], and Macdonald [8,Ch. 6].See also the author's previous paper [7].The same phenomena as described in the previous paragraph occur here, but in a very explicit form.See for instance Remarks 2.7 and 4.5 (referring to [9]) about the duality anti-algebra isomorphism and the shift operators, respectively.
In [7] I also discussed how the algebra AW (3, Q 0 ) is related to the double affine Hecke algebra (DAHA) of type (C ∨ 1 , C 1 ).In the basic (or polynomial) representation of this DAHA (denoted by H) on the space of Laurent polynomials in z, the nonsymmetric Askey-Wilson polynomials occur as eigenfunctions of a suitable element Y of H (see [13,9], [8,Ch. 6]).It turns out (see [7]) that a central extension AW (3, Q 0 ) of AW (3, Q 0 ) can be embedded as a subalgebra of H.As pointed out in the present paper (see Remark 2.7), the duality anti-algebra isomorphisms for AW (3, Q 0 ) and H are compatible with this embedding.
It would be interesting, also for possible generalisations to higher rank, to have a more conceptual way of decribing the relationship between Zhedanov's algebra and the double affine Hecke algebra.The present paper establishes this by showing that the algebra AW (3, Q 0 ) is isomorphic to the spherical subalgebra of H.The definition of a spherical subalgebra of a DAHA goes back to Etingof & Ginzburg [3], where a similar object was defined in the context of Cherednik algebras, see also [1].The definition of spherical subalgebra for the DAHA of type (C ∨ 1 , C 1 ) was given by Oblomkov [10].In general, the spherical subalgebra of a DAHA H is the algebra P sym HP sym , where P sym is the symmetrizer idempotent in H.
The proof of the isomorphism of AW (3, Q 0 ) with P sym HP sym in Section 3 is somewhat technical.It heavily uses the explicit relations given in [7] for the algebras AW (3, Q 0 ), AW (3, Q 0 ) and H.In Section 4 a similar isomorphism is proved between AW (3, Q 0 ) with two of its parameters q-shifted and P − sym HP − sym , where P − sym is the antisymmetrizing idempotent in H.In the final Section 5 it is shown as a corollary of these two isomorphisms that AW (3, Q 0 ) is isomorphic with the centralizer of T 1 in H.There it is also shown that the center of AW (3, Q 0 ) is trivial, with a proof in the same spirit as the proof of the faithfulness of the basic representation of AW (3, Q 0 ) given in [7].Combination of the various results finally gives as a corollary that the center of H is trivial.

Askey-Wilson polynomials
The Askey-Wilson polynomials are given by p n 1 2 (z + z −1 ); a, b, c, d | q := (ab, ac, ad; q) n a n 4 φ 3 q −n , q n−1 abcd, az, az −1 ab, ac, ad ; q, q (see [4] for the definition of q-shifted factorials (a; q) n and of q-hypergeometric series r φ s ).These polynomials are symmetric in a, b, c, d.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): The polynomials P n [z] are eigenfunctions of the operator D sym acting on the space A sym of symmetric Laurent polynomials f The eigenvalue equation is Under condition (1.1) all eigenvalues in (2.3) are distinct.

Zhedanov's algebra
Zhedanov's algebra AW (3) (see [15,5]) can in the q-case be described as an algebra with two generators K 0 , K 1 and with two relations Here the structure constants B, C 0 , C 1 , D 0 , D 1 are fixed complex constants.
There is a Casimir operator Q commuting with K 0 , K 1 : (2.6) 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: Then there is a representation (the basic representation or polynomial representation) of the algebra AW (3) with structure constants (2.7) on the space A sym of symmetric Laurent polynomials as follows: where D sym is the operator (2.2) having the Askey-Wilson polynomials as eigenfunctions.The Casimir operator Q becomes constant in this representation: where (Note the slight error in this formula in [7, (2.8)], version v3.It is corrected in v4.) Remark 2.1.Write AW (3) = AW (3; K 0 , K 1 ; a, b, c, d; q) in order to emphasize the dependence of AW (3) on the generators and the parameters (by (2.4), (2.5), (2.7)).There are several symmetries of this algebra.First of all it is invariant under permutations of a, b, c, d.The following one will be compatible with the duality of the double affine Hecke algebra to be discussed below: There is an anti-algebra isomorphism , ab , ac , ad Note that there is also a trivial anti-algebra isomorphism There is also an algebra isomorphism Let AW (3, Q 0 ) be the algebra generated by K 0 , K 1 with relations (2.4), (2.5) and Q = Q 0 , assuming the structure constants (2.7).Then the basic representation of AW (3) is also a representation of AW (3, Q 0 ).We have the following theorem (see [7, Theorem 2.2]): Theorem 2.2.The elements Note that the anti-algebra isomorphism (2.11) induces an anti-algebra isomorphisms for AW (3, Q 0 ).

The double affine Hecke algebra of type (C
One of the ways to describe the double affine Hecke algebra of type (C ∨ 1 , C 1 ), denoted by H, is as follows (see [7, Proposition 5.2]): (2.12) By adding the relations for T Z and T Z −1 and by combining the relations for T Y and T Y −1 we see that For the following theorem see Sahi [12] in the general rank case.In [7,Theorem 5.3] it is proved for the rank one case only.
Then it was proved in [7, Theorem 6.2, Corollary 6.3]: Theorem 2.6.AW (3, Q 0 ) has a basis consisting of There is a unique algebra isomorphism from The elements in the image commute with T 1 .
Remark 2.7.Write H = H(Y, Z, T 1 ; a, b, c, d; q) in order to emphasize the dependence of H on the generators and the parameters.Then there is an anti-algebra isomorphism , ab , ac , ad (q −1 abcd) a, b, c, d; q).Then, by a slight adaptation of (2.11), there is an anti-algebra isomorphism , ab , ac , ad ; q .
The two anti-algebra isomorphisms are compatible under the algebra embedding of AW (3, Q 0 ) into H given in Theorem 2.6.

The spherical subalgebra
From now on assume ab = 1.In H put Then In the basic representation of H we have for f ∈ A: Then Hence the image S( H) is a subalgebra of H.We call it the spherical subalgebra of H.
For U ∈ H we have in the basic representation: Hence, for the basic representation of H restricted to S( H), A sym is an invariant subspace.This representation of S( H) on A sym is faithful.Indeed, if S(U ) f = 0 for all f ∈ A sym then S(U ) f = 0 for all f ∈ A, so S(U ) = 0 by the faithfulness of the basic representation of H on A. Z H(T 1 ), the centralizer of T 1 in H, is a subalgebra of H.It has P sym as a central element.Hence So S restricted to Z H(T 1 ) is an algebra homomorphism.The algebra AW (3, Q 0 ) was defined by Definition 2.5.By Theorem 2.6 there is an algebra isomorphism from ).So we may consider AW (3, Q 0 ) as a subalgebra of Z H(T 1 ) and (3.1) will hold for U, V ∈ AW (3, Q 0 ).By (2.4), (2.5), (2.9) and (2.6), the algebra AW (3, Q 0 ) can be presented by the same generators and relations as for AW (3, Q 0 ) but with additional relation T 1 = −ab.
By Theorem 2.6 AW (3, Q 0 ) has a basis consisting of the elements By Theorem 2.2 AW (3, Q 0 ) has a basis consisting of Hence the map which sends a basis element ) extends linearly to a linear bijection from AW (3, Q 0 ) onto S( AW (3, Q 0 )).In fact, this map remains well-defined if we write it as where 3) can then be seen to be an algebra homomorphism.Indeed, consider the linear map U → U as a map to AW (3, Q 0 ) from the free algebra generated by K 0 , K 1 , T 1 with T 1 central such that it sends a word involving K 0 , K 1 , T 1 to the same word in AW (3, Q 0 ).This map is an algebra homomorphism.Composing it with S yields the map (3.3) which is again an algebra homomorphism.Now we have to check that R is sent to zero by the map (3.3) if R = 0 is a relation for AW (3, Q 0 ).This is clearly the case for R := T 1 + ab, since (T 1 + ab)(T 1 + 1) = 0.It is also clear for the other relations R = 0 in AW (3, Q 0 ) since these can be taken as the relations (2.14)-(2.16),which are also relations for AW (3, Q 0 ).So we have shown: For the proof note first that H has a basis consisting of the elements Hence S( H) is spanned by the elements Proof .The procedure will be as follows: 1. Write (T 1 + 1)Z m Y n (T 1 + 1) as a linear combination of This is done by induction, starting with the H relations for 2. Also write K m 1 K n 0 (T 1 + 1) and K m−1 (T 1 + 1) (m, n = 0, 1, . ..) as a linear combination of (3.5).
3. These latter linear combinations turn out to span the linear combinations obtained for (T 1 + 1)Z m Y n (T 1 + 1).

P
sym projects A onto A sym .Define the linear map S : H → H by S(U ) := P sym U P sym (U ∈ H).

Definition 3 . 3 .Lemma 3 . 4 .
Let m, n ∈ Z.For an element in H which is a linear combination of basis elements Z k Y l we say thatk,l∈Z c k,l Z k Y l = o(Z m Y n ) if c k,l = 0 implies |k| ≤ |m|, |l| ≤ |n|, (|k|, |l|) = (|m|, |n|).Theorem 3.2 will follow by induction with respect to |m| + |n| from the following lemma: Let m, n ∈ Z. Then