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 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(...

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Дата:	2008
Автор:	Koornwinder, T.H.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2008
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 15 назв. — англ.

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spelling	irk-123456789-1490292019-02-20T01:29:02Z Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra Koornwinder, T.H. This paper builds on the previous paper 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. 2008 Article Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 15 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33D80 http://dspace.nbuv.gov.ua/handle/123456789/149029 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	This paper builds on the previous paper 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.
format	Article
author	Koornwinder, T.H.
spellingShingle	Koornwinder, T.H. Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Koornwinder, T.H.
author_sort	Koornwinder, T.H.
title	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
title_short	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
title_full	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
title_fullStr	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
title_full_unstemmed	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
title_sort	zhedanov's algebra aw(3) and the double affine hecke algebra in the rank one case. ii. the spherical subalgebra
publisher	Інститут математики НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/149029
citation_txt	Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 15 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT koornwinderth zhedanovsalgebraaw3andthedoubleaffineheckealgebraintherankonecaseiithesphericalsubalgebra
first_indexed	2025-07-12T20:55:23Z
last_indexed	2025-07-12T20:55:23Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 052, 17 pages Zhedanov’s Algebra AW (3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra? Tom H. KOORNWINDER Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands E-mail: thk@science.uva.nl URL: http://www.science.uva.nl/∼thk/ Received November 15, 2007, in final form June 03, 2008; Published online June 10, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/052/ Abstract. This paper builds on the previous paper by the author, where a relationship between Zhedanov’s algebra AW (3) and the double affine Hecke algebra (DAHA) corre- sponding 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. Key words: Zhedanov’s algebra AW (3); double affine Hecke algebra in rank one; Askey– Wilson polynomials; spherical subalgebra 2000 Mathematics Subject Classification: 33D80 1 Introduction Zhedanov [15] introduced in 1991 an algebra AW (3) with three generators K0, K1, K2 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, K0 acts as the second order q-difference operator Dsym for which the Askey–Wilson polynomials are eigenfunctions and K1 acts as multiplication by z+z−1. The Casimir operator Q for AW (3) becomes a scalar Q0 in this representation. Let AW (3, Q0) be AW (3) with the additional relation Q = Q0. Then the basic representation AW (3, Q0) is faithful, see [7]. There is a parameter changing anti-algebra isomorphism of AW (3) which interchanges K0 and K1, and hence interchanges Dsym 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 Pn[z] as kernel which maps symmetric Laurent polynomials to infinite sequences {cn}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 rep- resentation 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 Mac- donald polynomials, and Macdonald’s conjectures for ordinary (symmetric) Macdonald polyno- mials associated with root systems could be proved. The idea of nonsymmetric polynomials ?This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at http://www.emis.de/journals/SIGMA/symmetry2007.html mailto:thk@science.uva.nl http://www.science.uva.nl/~thk/ http://www.emis.de/journals/SIGMA/2008/052/ http://www.emis.de/journals/SIGMA/symmetry2007.html 2 T.H. Koornwinder 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 , C1) (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, Q0) is related to the double affine Hecke algebra (DAHA) of type (C∨ 1 , C1). In the basic (or polynomial) representation of this DAHA (denoted by H̃) on the space of Laurent polynomials in z, the nonsymmetric Askey–Wilson polynomials occur as eigenfunctions of a suitable element Y of H̃ (see [13, 9], [8, Ch. 6]). It turns out (see [7]) that a central extension ÃW (3, Q0) of AW (3, Q0) can be embedded as a subalgebra of H̃. As pointed out in the present paper (see Remark 2.7), the duality anti-algebra isomorphisms for ÃW (3, Q0) and H̃ 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, Q0) is isomorphic to the spherical subalgebra of H̃. 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 , C1) was given by Oblomkov [10]. In general, the spherical subalgebra of a DAHA H̃ is the algebra PsymH̃Psym, where Psym is the symmetrizer idempotent in H̃. The proof of the isomorphism of AW (3, Q0) with PsymH̃Psym in Section 3 is somewhat tech- nical. It heavily uses the explicit relations given in [7] for the algebras AW (3, Q0), ÃW (3, Q0) and H̃. In Section 4 a similar isomorphism is proved between AW (3, Q0) with two of its parame- ters q-shifted and P− symH̃P− sym, where P− sym is the antisymmetrizing idempotent in H̃. In the final Section 5 it is shown as a corollary of these two isomorphisms that ÃW (3, Q0) is isomorphic with the centralizer of T1 in H̃. There it is also shown that the center of AW (3, Q0) is trivial, with a proof in the same spirit as the proof of the faithfulness of the basic representation of AW (3, Q0) given in [7]. Combination of the various results finally gives as a corollary that the center of H̃ is trivial. Conventions Throughout assume that q and a, b, c, d are complex constants such that q 6= 0, qm 6= 1 (m = 1, 2, . . .), a, b, c, d 6= 0, abcd 6= q−m (m = 0, 1, 2, . . .). (1.1) Let e1, e2, e3, e4 be the elementary symmetric polynomials in a, b, c, d: e1 := a + b + c + d, e2 := ab + ac + bc + ad + bd + cd, e3 := abc + abd + acd + bcd, e4 := abcd. (1.2) For Laurent polynomials f in z the z-dependence will be written as f [z]. Symmetric Laurent polynomials f [z] = ∑n k=−n ckz k (where ck = c−k) are related to ordinary polynomials f(x) in x = 1 2(z + z−1) by f(1 2(z + z−1)) = f [z]. Zhedanov’s Algebra AW (3). II 3 2 Summary of earlier results This section summarizes some results from [7], while Remarks 2.1 and 2.7 about the duality anti-algebra isomorphism are new here. 2.1 Askey–Wilson polynomials The Askey–Wilson polynomials are given by pn ( 1 2(z + z−1); a, b, c, d \| q ) := (ab, ac, ad; q)n an 4φ3 ( q−n, qn−1abcd, 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 rφs). 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 zn equals 1): Pn[z] = Pn[z; a, b, c, d \| q] := 1 (abcdqn−1; q)n pn ( 1 2(z + z−1); a, b, c, d \| q ) . (2.1) The polynomials Pn[z] are eigenfunctions of the operator Dsym acting on the space Asym of symmetric Laurent polynomials f [z] = f [z−1]: (Dsymf)[z] := (1− az)(1− bz)(1− cz)(1− dz) (1− z2)(1− qz2) ( f [qz]− f [z] ) + (a− z)(b− z)(c− z)(d− z) (1− z2)(q − z2) ( f [q−1z]− f [z] ) + (1 + q−1abcd)f [z]. (2.2) The eigenvalue equation is DsymPn = λnPn, λn := q−n + abcdqn−1. (2.3) Under condition (1.1) all eigenvalues in (2.3) are distinct. The three-term recurrence relation for the monic Askey–Wilson polynomials is (z + z−1)Pn[z] = Pn+1[z] + βnPn[z] + γnPn−1[z] (n = 0, 1, 2, . . .), where βn and γn are suitable constants and P−1[z] := 0 (see [6, (3.1.5)]). 2.2 Zhedanov’s algebra Zhedanov’s algebra AW (3) (see [15, 5]) can in the q-case be described as an algebra with two generators K0, K1 and with two relations (q + q−1)K1K0K1 −K2 1K0 −K0K 2 1 = B K1 + C0 K0 + D0, (2.4) (q + q−1)K0K1K0 −K2 0K1 −K1K 2 0 = B K0 + C1 K1 + D1. (2.5) Here the structure constants B, C0, C1, D0, D1 are fixed complex constants. There is a Casimir operator Q commuting with K0, K1: Q = K1K0K1K0 − (q2 + 1 + q−2)K0K1K0K1 + (q + q−1)K2 0K2 1 + (q + q−1)(C0K 2 0 + C1K 2 1 ) + B ( (q + 1 + q−1)K0K1 + K1K0 ) + (q + 1 + q−1)(D0K0 + D1K1). (2.6) 4 T.H. Koornwinder Let the structure constants be expressed in terms of a, b, c, d by means of e1, e2, e3, e4 (see (1.2)) as follows: B := (1− q−1)2(e3 + qe1), C0 := (q − q−1)2, C1 := q−1(q − q−1)2e4, (2.7) D0 := −q−3(1− q)2(1 + q)(e4 + qe2 + q2), D1 := −q−3(1− q)2(1 + q)(e1e4 + qe3). Then there is a representation (the basic representation or polynomial representation) of the alge- bra AW (3) with structure constants (2.7) on the space Asym of symmetric Laurent polynomials as follows: (K0f)[z] := (Dsymf)[z], (K1f)[z] := (Z + Z−1)f [z] := (z + z−1)f [z], (2.8) where Dsym is the operator (2.2) having the Askey–Wilson polynomials as eigenfunctions. The Casimir operator Q becomes constant in this representation: (Qf)(z) = Q0 f(z), (2.9) where Q0 := q−4(1− q)2 ( q4(e4 − e2) + q3(e2 1 − e1e3 − 2e2) − q2(e2e4 + 2e4 + e2) + q(e2 3 − 2e2e4 − e1e3) + e4(1− e2) ) . (2.10) (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;K0,K1; 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 AW (3;K0,K1; a, b, c, d; q) → AW ( 3; aK1, (q−1abcd)− 1 2 K0; 1 (q−1abcd) 1 2 , ab (q−1abcd) 1 2 , ac (q−1abcd) 1 2 , ad (q−1abcd) 1 2 ; q ) . (2.11) See [15, § 2]. Under this mapping Q is sent to qa(bcd)−1Q and Q0 to qa(bcd)−1Q0. Note that there is also a trivial anti-algebra isomorphism AW (3;K0,K1; a, b, c, d; q) → AW (3;K0,K1; a, b, c, d; q). This sends Q to Q and Q0 to Q0. There is also an algebra isomorphism AW (3;K0,K1; a, b, c, d; q) → AW ( 3; q abcd K0,K1; a−1, b−1, c−1, d−1; q−1 ) . This sends Q to q2(abcd)−2Q and Q0 to q2(abcd)−2Q0. Zhedanov’s Algebra AW (3). II 5 Let AW (3, Q0) be the algebra generated by K0,K1 with relations (2.4), (2.5) and Q = Q0, assuming the structure constants (2.7). Then the basic representation of AW (3) is also a repre- sentation of AW (3, Q0). We have the following theorem (see [7, Theorem 2.2]): Theorem 2.2. The elements Kn 0 (K1K0)lKm 1 (m,n = 0, 1, 2, . . . , l = 0, 1) form a basis of AW (3, Q0) and the representation (2.8) of AW (3, Q0) is faithful. Note that the anti-algebra isomorphism (2.11) induces an anti-algebra isomorphisms for AW (3, Q0). 2.3 The double affine Hecke algebra of type (C∨ 1 , C1) One of the ways to describe the double affine Hecke algebra of type (C∨ 1 , C1), denoted by H̃, is as follows (see [7, Proposition 5.2]): Definition 2.3. H̃ is the algebra generated by T1, Y , Y −1, Z, Z−1 with relations Y Y −1 = 1 = Y −1Y , ZZ−1 = 1 = Z−1Z and T 2 1 = −(ab + 1)T1 − ab, T1Z = Z−1T1 + (ab + 1)Z−1 − (a + b), T1Z −1 = ZT1 − (ab + 1)Z−1 + (a + b), T1Y = q−1abcdY −1T1 − (ab + 1)Y + ab(1 + q−1cd), T1Y −1 = q(abcd)−1Y T1 + q(abcd)−1(1 + ab)Y − q(cd)−1(1 + q−1cd), Y Z = qZY + (1 + ab)cd Z−1Y −1T1 − (a + b)cd Y −1T1 − (1 + q−1cd)Z−1T1 − (1− q)(1 + ab)(1 + q−1cd)Z−1 + (c + d)T1 + (1− q)(a + b)(1 + q−1cd), Y Z−1 = q−1Z−1Y − q−2(1 + ab)cd Z−1Y −1T1 + q−2(a + b)cd Y −1T1 + q−1(1 + q−1cd)Z−1T1 − q−1(c + d)T1, Y −1Z = q−1ZY −1 − q(ab)−1(1 + ab)Z−1Y −1T1 + (ab)−1(a + b)Y −1T1 + q(abcd)−1(1 + q−1cd)Z−1T1 + q(abcd)−1(1− q)(1 + ab)(1 + q−1cd)Z−1 − (abcd)−1(c + d)T1 − (abcd)−1(1− q)(1 + ab)(c + d), Y −1Z−1 = qZ−1Y −1 + q(ab)−1(1 + ab)Z−1Y −1T1 − (ab)−1(a + b)Y −1T1 − q2(abcd)−1(1 + q−1cd)Z−1T1 + q(abcd)−1(c + d)T1. (2.12) By adding the relations for TZ and TZ−1 and by combining the relations for TY and TY −1 we see that T1(Z + Z−1) = (Z + Z−1)T1, T1(Y + q−1abcdY −1) = (Y + q−1abcdY −1)T1. 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. Theorem 2.4. A basis of H̃ is provided by the elements ZmY nT i 1, where m, n ∈ Z, i = 0, 1. The basic or polynomial representation of H̃ is a representation of H̃ on the space A of Laurent polynomials f [z] in z such that (Zf)[z] := z f [z] and T1 and Y act as q-difference-reflection operators given by [7, (3.11), (3.13)]. This representation is faithful. If ab 6= 1 then T1 acting 6 T.H. Koornwinder on A has eigenspaces Asym (for eigenvalue −ab) and A−sym (for eigenvalue −1), and A is the direct sum of Asym and A−sym. In the basic representation of H̃ the operator Y + q−1abcdY −1 has eigenvalues λn (see (2.3)), of multiplicity 2 for n = 1, 2, . . . and of multiplicity 1 for n = 0. If n ≥ 1 and ab 6= 1 then the eigenspace of λn splits as a one-dimensional part in Asym spanned by the Askey–Wilson polynomial Pn[z] given by (2.1) and a one-dimensional part in A−sym spanned by Qn[z] := a−1b−1z−1(1− az)(1− bz) Pn−1[z; qa, qb, c, d \| q]. (2.13) In [7, § 6] the following algebra ÃW (3, Q0) was defined, which is a central extension of AW (3, Q0): Definition 2.5. The algebra ÃW (3, Q0) is generated by K0, K1, T1 such that (T1+ab)(T1+1) = 0, T1 commutes with K0, K1, and with further relations (q + q−1)K1K0K1 −K2 1K0 −K0K 2 1 = B K1 + C0 K0 + D0 + E K1(T1 + ab) + F0(T1 + ab), (2.14) (q + q−1)K0K1K0 −K2 0K1 −K1K 2 0 = B K0 + C1 K1 + D1 + E K0(T1 + ab) + F1(T1 + ab), (2.15) Q0 = (K1K0)2 − (q2 + 1 + q−2)K0(K1K0)K1 + (q + q−1)K2 0K2 1 + (q + q−1)(C0K 2 0 + C1K 2 1 ) + ( B + E(T1 + ab) )( (q + 1 + q−1)K0K1 + K1K0 ) + (q + 1 + q−1) ( D0 + F0(T1 + ab) ) K0 + (q + 1 + q−1) ( D1 + F1(T1 + ab) ) K1 + G(T1 + ab). (2.16) Here the structure constants are given by (2.7) together with E := −q−2(1− q)3(c + d), F0 := q−3(1− q)3(1 + q)(cd + q), F1 := q−3(1− q)3(1 + q)(a + b)cd, G := −q−4(1− q)3 ( (a + b)(c + d) ( cd(q2 + 1) + q ) − q(ab + 1) ( (c2 + d2)(q + 1)− cd ) + (cd + e4)(q2 + 1) + (e2 + e4 − ab)q3 ) , and Q0 is given by (2.10). Then it was proved in [7, Theorem 6.2, Corollary 6.3]: Theorem 2.6. ÃW (3, Q0) has a basis consisting of Kn 0 (K1K0)iKm 1 T j 1 (m,n = 0, 1, 2, . . . , i, j = 0, 1). There is a unique algebra isomorphism from ÃW (3, Q0) into H̃ such that K0 7→ Y +q−1abcdY −1, K1 7→ Z + Z−1, T1 7→ T1. The elements in the image commute with T1. Remark 2.7. Write H̃ = H̃(Y, Z, T1; a, b, c, d; q) in order to emphasize the dependence of H̃ on the generators and the parameters. Then there is an anti-algebra isomorphism H̃(Y, Z, T1; a, b, c, d; q) → H̃ ( aZ−1, (q−1abcd)− 1 2 Y −1, T1; 1 (q−1abcd) 1 2 , ab (q−1abcd) 1 2 , ac (q−1abcd) 1 2 , ad (q−1abcd) 1 2 ; q ) , Zhedanov’s Algebra AW (3). II 7 see [9, Proposition 8.5(i)]. Also write ÃW (3, Q0) = ÃW (3, Q0;K0,K1, T1; a, b, c, d; q). Then, by a slight adaptation of (2.11), there is an anti-algebra isomorphism ÃW (3, Q0;K0,K1, T1; a, b, c, d; q) → ÃW ( 3; qa(bcd)−1Q0; aK1, (q−1abcd)− 1 2 K0; 1 (q−1abcd) 1 2 , ab (q−1abcd) 1 2 , ac (q−1abcd) 1 2 , ad (q−1abcd) 1 2 ; q ) . The two anti-algebra isomorphisms are compatible under the algebra embedding of ÃW (3, Q0) into H̃ given in Theorem 2.6. 3 The spherical subalgebra From now on assume ab 6= 1. In H̃ put Psym := (1− ab)−1(T1 + 1). Then P 2 sym = Psym. In the basic representation of H̃ we have for f ∈ A: Psymf = { f if T1f = −abf , 0 if T1f = −f . Psym projects A onto Asym. Define the linear map S : H̃ → H̃ by S(U) := PsymUPsym (U ∈ H̃). Then S(U) S(V ) = S(UPsymV ) (U, V ∈ H̃). Hence the image S(H̃) is a subalgebra of H̃. We call it the spherical subalgebra of H̃. For U ∈ H̃ we have in the basic representation: S(U) f = PsymUPsymf = { PsymU f if T1f = −abf , 0 if T1f = −f . Hence, for the basic representation of H̃ restricted to S(H̃), Asym is an invariant subspace. This representation of S(H̃) on Asym is faithful. Indeed, if S(U) f = 0 for all f ∈ Asym then S(U) f = 0 for all f ∈ A, so S(U) = 0 by the faithfulness of the basic representation of H̃ on A. ZH̃(T1), the centralizer of T1 in H̃, is a subalgebra of H̃. It has Psym as a central element. Hence S(U) = UPsym and S(UV ) = S(U) S(V ) (U, V ∈ ZH̃(T1)). (3.1) So S restricted to ZH̃(T1) is an algebra homomorphism. The algebra ÃW (3, Q0) was defined by Definition 2.5. By Theorem 2.6 there is an algeb- ra isomorphism from ÃW (3, Q0) into H̃ such that K0 7→ Y + q−1abcdY −1, K1 7→ Z + Z−1, 8 T.H. Koornwinder T1 7→ T1. This isomorphism embeds ÃW (3, Q0) into ZH̃(T1). So we may consider ÃW (3, Q0) as a subalgebra of ZH̃(T1) and (3.1) will hold for U, V ∈ ÃW (3, Q0). By (2.4), (2.5), (2.9) and (2.6), the algebra AW (3, Q0) can be presented by the same genera- tors and relations as for ÃW (3, Q0) but with additional relation T1 = −ab. By Theorem 2.6 ÃW (3, Q0) has a basis consisting of the elements Kn 0 (K1K0)iKm 1 (T1 + 1), Kn 0 (K1K0)iKm 1 (T1 + ab) (m,n = 0, 1, 2, . . . , i = 0, 1). Hence S(ÃW (3, Q0)) has a basis consisting of (1− ab)−1Kn 0 (K1K0)iKm 1 (T1 + 1) (m,n = 0, 1, 2, . . . , i = 0, 1). By Theorem 2.2 AW (3, Q0) has a basis consisting of Kn 0 (K1K0)iKm 1 (m,n = 0, 1, 2, . . . , i = 0, 1). (3.2) Hence the map which sends a basis element Kn 0 (K1K0)iKm 1 of AW (3, Q0) to a basis element (1 − ab)−1Kn 0 (K1K0)iKm 1 (T1 + 1) of S(ÃW (3, Q0)) extends linearly to a linear bijection from AW (3, Q0) onto S(ÃW (3, Q0)). In fact, this map remains well-defined if we write it as U 7→ (1− ab)−1Ũ(T1 + 1), (3.3) where U 7→ Ũ sends words U in AW (3, Q0) to corresponding words Ũ in ÃW (3, Q0). Moreover, the map (3.3) can then be seen to be an algebra homomorphism. Indeed, consider the linear map U 7→ Ũ as a map to ÃW (3, Q0) from the free algebra generated by K0, K1, T1 with T1 central such that it sends a word involving K0, K1, T1 to the same word in ÃW (3, Q0). 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, Q0). This is clearly the case for R := T1 +ab, since (T1 +ab)(T1 +1) = 0. It is also clear for the other relations R = 0 in AW (3, Q0) since these can be taken as the relations (2.14)–(2.16), which are also relations for ÃW (3, Q0). So we have shown: Proposition 3.1. The map (3.3), where U 7→ Ũ sends words U involving K0,K1 in AW (3, Q0) to the same words Ũ in ÃW (3, Q0), is a well defined algebra isomorphism from AW (3, Q0) onto S(ÃW (3, Q0)). Theorem 3.2. S(H̃) = S(ÃW (3, Q0)), so the spherical subalgebra S(H̃) is isomorphic to the algebra AW (3, Q0) by the map (3.3) sending AW (3, Q0) to S(H̃). For the proof note first that H̃ has a basis consisting of the elements ZmY n(T1 + 1), ZmY n(T1 + ab) (m,n ∈ Z). (3.4) Hence S(H̃) is spanned by the elements (T1 + 1)ZmY n(T1 + 1) (m,n ∈ Z). Definition 3.3. Let m,n ∈ Z. For an element in H̃ which is a linear combination of basis elements ZkY l we say that∑ k,l∈Z ck,lZ kY l = o(ZmY n) if ck,l 6= 0 implies \|k\| ≤ \|m\|, \|l\| ≤ \|n\|, (\|k\|, \|l\|) 6= (\|m\|, \|n\|). Zhedanov’s Algebra AW (3). II 9 Theorem 3.2 will follow by induction with respect to \|m\|+ \|n\| from the following lemma: Lemma 3.4. Let m,n ∈ Z. Then (T1 + 1)ZmY n(T1 + 1) ∈ (T1 + 1) ( ÃW (3, Q0) + o(ZmY n) ) (T1 + 1). Proof. The procedure will be as follows: 1. Write (T1 + 1)ZmY n(T1 + 1) as a linear combination of Z \|m\|Y \|n\|(T1 + 1), Z−\|m\|Y \|n\|(T1 + 1), Z \|m\|Y −\|n\|(T1 + 1), Z−\|m\|Y −\|n\|(T1 + 1) (mod o(Z \|m\|Y \|n\|)(T1 + 1)). (3.5) This is done by induction, starting with the H̃ relations for T1Z, T1Z −1, T1Y , T1Y −1. 2. Also write Km 1 Kn 0 (T1 + 1) and Km−1 1 K0K1K n−1 0 (T1 + 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 (T1 + 1)ZmY n(T1 + 1). Step 1. We get the following expressions in terms of the elements (3.5) (here m,n ∈ {1, 2, . . .}) (T1 + 1)Zm(T1 + 1) = ( Zm + Z−m + o(Zm) ) (T1 + 1), (3.6) (T1 + 1)Z−m(T1 + 1) = −ab ( Zm + Z−m + o(Zm) ) (T1 + 1), (3.7) (T1 + 1)Y n(T1 + 1) = −ab ( Y n + (q−1abcd)nY −n + o(Y n) ) (T1 + 1), (3.8) (T1 + 1)Y −n(T1 + 1) = ( (q−1abcd)−nY n + Y −n + o(Y n) ) (T1 + 1), (3.9) (T1 + 1)ZmY n(T1 + 1) = ( ZmY n − ab(q−1abcd)nZ−mY −n + o(ZmY n) ) (T1 + 1), (3.10) (T1 + 1)Z−mY n(T1 + 1) = ( −(ab + 1)ZmY n − ab(q−1abcd)nZmY −n − abZ−mY n + o(ZmY n) ) (T1 + 1), (3.11) (T1 + 1)ZmY −n(T1 + 1) = ( ZmY −n + (q−1abcd)−nZ−mY n + (1 + ab)Z−mY −n + o(ZmY n) ) (T1 + 1), (3.12) (T1 + 1)Z−mY −n(T1 + 1) = ( (q−1abcd)−nZmY n − abZ−mY −n + o(ZmY n) ) (T1 + 1). (3.13) Step 2. We get the following expressions in terms of the elements (3.5) (here m,n ∈ {1, 2, . . .}) Km 1 (T1 + 1) = ( Zm + Z−m + o(Zm) ) (T1 + 1), (3.14) Kn 0 (T1 + 1) = ( Y n + (q−1abcd)nY −n + o(Y n) ) (T1 + 1), (3.15) Km 1 Kn 0 (T1 + 1) = ( ZmY n + Z−mY n + (q−1abcd)n ( ZmY −n + Z−mY −n ) + o(ZmY n) ) (T1 + 1), (3.16) Km−1 1 K0K1K n−1 0 (T1 + 1) = ( qZmY n + q−1Z−mY n + q−1(q−1abcd)nZmY −n + q−1(q−1abcd)n(1 + ab− q2ab)Z−mY −n + o(ZmY n) ) (T1 + 1). (3.17) 10 T.H. Koornwinder Step 3. The only cases which may not be immediately clear are for (T1 + 1)Z±mY ±n(T1 + 1) (m,n ∈ {1, 2, . . .}). Then we have by comparing the identities in Step 1 and Step 2: (T1 + 1)ZmY n(T1 + 1) = 1 (1− q2) ( Km−1 1 (K1K0 − qK0K1)Kn−1 0 + o(ZmY n) ) (T1 + 1), (T1 + 1)Z−mY n(T1 + 1) = q (1− q2) ( Km−1 1 (−q−1 ( 1 + ab− q2ab)K1K0 + K0K1 ) Kn−1 0 + o(ZmY n) ) (T1 + 1), (T1 + 1)ZmY −n(T1 + 1) = q (1− q2)(q−1abcd)n ( Km−1 1 (−qK1K0 + K0K1 ) Kn−1 0 + o(ZmY n) ) (T1 + 1), (T1 + 1)Z−mY −n(T1 + 1) = 1 (1− q2)(q−1abcd)n ( Km−1 1 (K1K0 − qK0K1 ) Kn−1 0 + o(ZmY n) ) (T1 + 1). Proofs for Step 1. We will use repeatedly that (T1 +1)2 = (1−ab)(T1 +1). From the relation for T1Z in (2.12) we obtain (T1 + 1)Z = Z−1(T1 + 1) + Z + abZ−1 − (a + b), (3.18) (T1 + 1)Z(T1 + 1) = ( Z + Z−1 − (a + b) ) (T1 + 1) = ( Z + Z−1 + o(Z) ) (T1 + 1), i.e., (3.6) for m = 1. Now we will prove (3.6) by induction. Suppose it holds for some positive integer m, then we will prove the identity with m replaced by m + 1. By successively substitu- ting (3.18) and the induction hypothesis we have: (T1 + 1)Zm+1(T1 + 1) = Z−1(T1 + 1)Zm(T1 + 1) + ( Zm+1 + o(Zm+1) ) (T1 + 1) = Z−1 ( Zm + Z−m + o(Zm) ) (T1 + 1) + ( Zm+1 + o(Zm+1) ) (T1 + 1) = ( Zm+1 + Z−m−1 + o(Zm+1) ) (T1 + 1). Similarly, from the relation for T1Z −1 in (2.12) we prove (3.7) by induction: (T1 + 1)Z−1 = Z(T1 + 1)− Z − abZ−1 + (a + b), hence (T1 + 1)Z−1(T1 + 1) = ( −ab(Z + Z−1) + (a + b) ) (T1 + 1) = −ab ( Z + Z−1 + o(Z) ) (T1 + 1), (T1 + 1)Z−m−1(T1 + 1) = Z(T1 + 1)Z−m(T1 + 1)− ab ( Z−m−1 + o(Zm+1) ) (T1 + 1) = −abZ ( Zm + Z−m + o(Zm) ) (T1 + 1)− ab ( Z−m−1 + o(Zm+1) ) (T1 + 1) = −ab ( Zm+1 + Z−m−1 + o(Zm+1) ) (T1 + 1). The proofs of (3.8), (3.9) are similar. The proof of (3.10) is for fixed n by induction with respect to m. First we prove the case m = 1. By (3.18) and (3.8) we obtain: (T1 + 1)ZY n(T1 + 1) = ( Z−1(T1 + 1)Y n + ZY n + abZ−1Y n + o(ZY n) ) (T1 + 1) = ( ZY n − ab(q−1abcd)nZ−1Y −n + o(ZY n) ) (T1 + 1). Now suppose (3.10) holds for some positive integer m, then we will prove the identity with m replaced by m + 1. By (3.18) and the induction hypothesis we obtain: (T1 + 1)Zm+1Y n(T1 + 1) = ( Z−1(T1 + 1)ZmY n + Zm+1Y n + o(Zm+1Y n) ) (T1 + 1) = ( Zm+1Y n− ab(q−1abcd)nZ−m−1Y −n+ o(Zm+1Y n) ) (T1 + 1). Zhedanov’s Algebra AW (3). II 11 The proofs of (3.11), (3.12), (3.13) are similar. Proofs for Step 2. Formulas (3.14), (3.15), (3.16) immediately follow by the substitutions K0 = Y + q−1abcdY −1, K1 = Z + Z−1. As for (3.17) we first verify it for m = n = 1 by writing the left-hand side as (1 − ab)−1(T1 + 1)(Y + q−1abcdY −1)(Z + Z−1)(T1 + 1) and by using the last four relations in (2.12). Then we obtain for the case of general m, n: Km−1 1 K0K1K n−1 0 (T1 + 1) = (1− ab)−1Km−1 1 K0K1(T1 + 1)Kn−1 0 (T1 + 1) = (Z + Z−1)m−1 ( qZY + q−1Z−1Y + q−1(q−1abcd)ZY −1 + q−1(q−1abcd)(1 + ab− q2ab)Z−1Y −1 + o(ZY ) ) (Y + q−1abcdY −1)n−1(T1 + 1) = ( qZmY n + q−1Z−mY n + q−1(q−1abcd)nZmY −n + q−1(q−1abcd)n(1 + ab− q2ab)Z−mY −n + o(ZmY n) ) (T1 + 1). � 4 The antispherical subalgebra In H̃ put P− sym := (ab− 1)−1(T1 + ab). Then (P− sym)2 = P− sym. In the basic representation of H̃ we have for f ∈ A: P− symf = { f if T1f = −f , 0 if T1f = −abf . Let A−sym denote the eigenspace of T1 acting on A for eigenvalue −1. Then P− sym projects A onto A−sym. Define the linear map S− : H̃ → H̃ by S−(U) := P− symUP− sym (U ∈ H̃). Then S−(U) S−(V ) = S−(UP− symV ) (U, V ∈ H̃). Hence the image S−(H̃) is a subalgebra of H̃. We call it the antispherical subalgebra of H̃. For U ∈ H̃ we have in the basic representation: S−(U) f = P− symUP− symf = { P− symU f if T1f = −f , 0 if T1f = −abf . Hence, for the basic representation of H̃ restricted to S−(H̃), A−sym is an invariant subspace. Recall the algebra isomorphism from ÃW (3, Q0) into ZH̃(T1) given by Theorem 2.6. Since S−(U) S−(V ) = S−(UV ) for U, V ∈ ZH̃(T1), we see that ÃW (3, Q0), considered as a subalgabra of H̃ and hence of ZH̃(T1), is mapped by S− onto a subalgebra S−(ÃW (3, Q0)) of S−(H̃). From the basis (3.2) of ÃW (3, Q0) we see that S−(ÃW (3, Q0)) has basis (ab− 1)−1Kn 0 (K1K0)iKm 1 (T1 + ab) (m,n = 0, 1, 2, . . . , i = 0, 1). 12 T.H. Koornwinder Compare this basis with the basis (3.2) of AW (3, Q0). Thus the map which sends a basis element Kn 0 (K1K0)iKm 1 of AW (3, Q0) to a basis element qn+i(ab−1)−1Kn 0 (K1K0)iKm 1 (T1 +ab) of S−(ÃW (3, Q0)) extends linearly to a linear bijection from AW (3, Q0) onto S−(ÃW (3, Q0)). In fact this map extends to an algebra isomorphism: Proposition 4.1. Let AW (3, Q0; qa, qb, c, d) be AW (3, Q0) with a, b replaced by qa, qb, re- spectively. Then there is a well-defined algebra isomorphism from AW (3, Q0; qa, qb, c, d) onto S−(ÃW (3, Q0)) given by the map U 7→ (ab− 1)−1Ũ(T1 + ab), (4.1) where U 7→ Ũ sends words U in AW (3, Q0; qa, qb, c, d) involving K0, K1 to corresponding words Ũ in ÃW (3, Q0) involving qK0, K1. Proof. Consider the linear map U 7→ Ũ as a map to ÃW (3, Q0) from the free algebra generated by K0, K1, T1 with T1 central such that it sends a word involving K0, K1, T1 to the corresponding word involving qK0, K1, T1 in ÃW (3, Q0). This map is an algebra homomorphism. Composing it with S− yields the map (4.1) which is again an algebra homomorphism. Now we have to check that R is sent to zero by the map (4.1) if R = 0 is a relation for AW (3, Q0; qa, qb, c, d). This is clearly the case for R := T1 + 1, since (T1 + 1)(T1 + ab) = 0. To see this for the other relations, rewrite relations (2.14)–(2.16) for ÃW (3, Q0) as: (q + q−1)K1K0K1 −K2 1K0 −K0K 2 1 − ( B + (ab− 1)E ) K1 − C0 K0 − ( D0 + (ab− 1)F0 ) − (E K1 + F0)(T1 + 1) = 0, (q + q−1)K0K1K0 −K2 0K1 −K1K 2 0 − ( B + (ab− 1)E ) K0 − C1 K1 − ( D1 + (ab− 1)F1 ) − (E K0 + F1)(T1 + 1) = 0, K1K0K1K0 − (q2 + 1 + q−2)K0K1K0K1 + (q + q−1)K2 0K2 1 − ((q + q−1)(C0K 2 0 + C1K 2 1 ) + ( B + (ab− 1)E )( (q + 1 + q−1)K0K1+ K1K0 ) + (q + 1 + q−1) ( D0+ (ab− 1)F0 ) K0 + (q + 1 + q−1) ( D1 + (ab− 1)F1 ) K1 + (ab− 1)G−Q0 + (E K1K0 + G)(T1 + 1) +(q + 1 + q−1)(E K0K1 + F0 K0 + F1 K1)(T1 + 1) = 0. On multiplication with T1 + ab we see that the identities Ri(T1 + ab) = 0 (i = 1, 2, 3) must be valid in S−(ÃW (3, Q0)), where R1 := (q + q−1)K1K0K1 −K2 1K0 −K0K 2 1 − ( B + (ab− 1)E ) K1 − C0 K0 −D0 − (ab− 1)F0, R2 := (q + q−1)K0K1K0 −K2 0K1 −K1K 2 0 − (B + (ab− 1)E ) K0 − C1 K1 −D1 − (ab− 1)F1, R3 := K1K0K1K0 − (q2 + 1 + q−2)K0K1K0K1 + (q + q−1)K2 0K2 1 − ((q + q−1)(C0K 2 0 + C1K 2 1 ) + ( B + (ab− 1)E )( (q + 1 + q−1)K0K1 + K1K0 ) + (q + 1 + q−1) ( D0 + (ab− 1)F0 ) K0 + (q + 1 + q−1) ( D1 + (ab− 1)F1 ) K1 + (ab− 1)G−Q0. Now consider relations (2.4), (2.5) en (2.9) for AW (3, Q0; qa, qb, c, d) (so with a, b in the structure constants replaced by qa, qb). These can be written in the form Ui = 0 (i = 1, 2, 3) by bringing everything to the left-hand side in the relations. Now consider Ui as elements of the free algebra generated by K0, K1. For the images under the map (4.1) we then obtain the following elements of S−(ÃW (3, Q0)): Ũ1 = (1− ab)−1qR1, Ũ2 = (1− ab)−1q2R2, Ũ3 = (1− ab)−1q2R3, which are all zero. � Zhedanov’s Algebra AW (3). II 13 Theorem 4.2. S−(H̃) = S(ÃW (3, Q0)), so the antispherical subalgebra S−(H̃) is isomorphic to the algebra AW (3, Q0; qa, qb, c, d) by the map (4.1) sending AW (3, Q0; qa, qb, c, d) to S−(H̃). The proof is analogous to the proof of Theorem 3.2. Since H̃ has a basis (3.4), S−(H̃) is spanned by the elements (T1 + ab)ZmY n(T1 + ab) (m,n ∈ Z). Recall Definition 3.3. Theorem 4.2 will follow by induction with respect to \|m\| + \|n\| from the following analogue of Lemma 3.4: Lemma 4.3. Let m,n ∈ Z. Then (T1 + ab)ZmY n(T1 + ab) ∈ (T1 + ab) ( ÃW (3, Q0) + o(ZmY n) ) (T1 + ab). Proof. We use the same procedure as in the proof of Lemma 3.4. I will only list the main formulas in the three steps. The reader can verify these formulas in an analogous way as in the proof of Lemma 3.4. Step 1 (T1 + ab)Zm(T1 + ab) = ab ( Zm + Z−m + o(Zm) ) (T1 + ab), (T1 + ab)Z−m(T1 + ab) = − ( Zm + Z−m + o(Zm) ) (T1 + ab), (T1 + ab)Y n(T1 + ab) = − ( Y n + (q−1abcd)nY −n + o(Y n) ) (T1 + ab), (T1 + ab)Y −n(T1 + ab) = ab ( (q−1abcd)−nY n + Y −n + o(Y n) ) (T1 + ab), (T1 + ab)ZmY n(T1 + ab) = ( abZmY n − (q−1abcd)nZ−mY −n + o(ZmY n) ) (T1 + ab), (T1 + ab)Z−mY n(T1 + ab) = ( −(ab + 1)ZmY n − (q−1abcd)nZmY −n − Z−mY n + o(ZmY n) ) (T1 + ab), (T1 + ab)ZmY −n(T1 + ab) = ( ab(q−1abcd)−nZ−mY n + abZmY −n + (1 + ab)Z−mY −n + o(ZmY n) ) (T1 + ab), (T1 + ab)Z−mY −n(T1 + ab) = ( ab(q−1abcd)−nZmY n − Z−mY −n + o(ZmY n) ) (T1 + ab). Step 2 Km 1 (T1 + ab) = ( Zm + Z−m + o(Zm) ) (T1 + ab), Kn 0 (T1 + ab) = ( Y n + (q−1abcd)nY −n + o(Y n) ) (T1 + ab), Km 1 Kn 0 (T1 + ab) = ( ZmY n + Z−mY n + (q−1abcd)n ( ZmY −n + Z−mY −n ) + o(ZmY n) ) (T1 + ab), Km−1 1 K0K1K n−1 0 (T1 + ab) = ( qZmY n + q−1Z−mY n + q−1(q−1abcd)nZmY −n + (qab)−1(q−1abcd)n(1 + ab− q2)Z−mY −n + o(ZmY n) ) (T1 + ab). 14 T.H. Koornwinder Step 3 (T1 + ab)ZmY n(T1 + ab) = ab 1− q2 ( Km−1 1 (K1K0 − qK0K1)Kn−1 0 + o(ZmY n) ) (T1 + ab), (T1 + ab)Z−mY n(T1 + ab) = 1 1− q2 ( Km−1 1 (− ( 1 + ab− q2)K1K0 + qabK0K1 ) Kn−1 0 + o(ZmY n) ) (T1 + ab), (T1 + ab)ZmY −n(T1 + ab) = qab (1− q2)(q−1abcd)n ( Km−1 1 (−qK1K0 + K0K1 ) Kn−1 0 + o(ZmY n) ) (T1 + ab), (T1 + ab)Z−mY −n(T1 + ab) = ab (1− q2)(q−1abcd)n ( Km−1 1 (K1K0 − qK0K1 ) Kn−1 0 + o(ZmY n) ) (T1 + ab). � Remark 4.4. The same q-shift for the parameters as in Proposition 4.1 occurs in [7, (3.17)] (the eigenfunction of Y + q−1abcdY −1 in A−sym expressed in terms of Askey–Wilson polynomials). Of course, these two results are very much related to each other. Let H̃(qa, qb, c, d) be H̃ with parameters a, b, c, d replaced by qa, qb, c, d, respectively. If we compare Theorems 3.2 and 4.2 then we can conclude that the spherical subalgebras S−(H̃) and S(H̃(qa, qb, c, d)) are isomorphic. This result is an analogue of the result in [1, Proposition 4.11] for Cherednik algebras. Remark 4.5. Just as we had in Step 1 of the proofs of Lemma 3.4 and Lemma 4.3, we can derive from the first, fourth and fifth relation in (2.12) that (T1 + 1)(Y + q−1a2b2cdY −1 − (q−1abcd + ab))(T1 + 1) = 0, (T1 + ab)(Y + q−1cdY −1 − (q−1cd + 1))(T1 + ab) = 0. It follows that, in the basic representation of H̃, the operator D− := Y + q−1a2b2cdY −1 − (q−1abcd + ab) maps Asym into A−sym, while D+ := Y + q−1cdY −1 − (q−1cd + 1) maps A−sym into Asym. Since both operators preserve the eigenspace of λn, we obtain D− Pn = ab(q−1cd− q−n)(qn − 1) Qn, D+ Qn = (q−1cd− q−na−1b−1)(qnab− 1) Pn, where the Askey–Wilson polynomial Pn and the shifted Askey–Wilson polynomial Qn are given by (2.1) and (2.13), respectively, and the constant factors in the above identities follow by com- paring coefficients of zn. Thus the operators D− and D+ can be considered as shift operators. The observations in this Remark were earlier made (in different notation) in [9, Lemma 12.2, Proposition 12.3]. Zhedanov’s Algebra AW (3). II 15 5 Centralizers and centers As a corollary of Theorem 3.2 and Theorem 4.2 we obtain: Theorem 5.1. The centralizer ZH̃(T1) is equal to ÃW (3, Q0). Proof. Write U ∈ H̃ as U = (1− ab)−1U(T1 + 1) + (ab− 1)−1U(T1 + ab). (5.1) Suppose that U ∈ ZH̃(T1). Then U(T1 + 1) = (1− ab)−1(T1 + 1)U(T1 + 1), U(T1 + ab) = (ab− 1)−1(T1 + ab)U(T1 + ab). So U(T1+1) ∈ S(H̃) = S(ÃW (3, Q0))⊂ÃW (3, Q0) and U(T1+ab) ∈ S−(H̃) = S−(ÃW (3, Q0))⊂ ÃW (3, Q0). � The following theorem is interesting in its own right, but it can also be used, in combination with Theorems 3.2 and 4.2, in order to show that the center of H̃ consists of the scalars (see Theorem 5.3). The proof is in the same spirit as the proof of the faithfulness of the basic representation of AW (3, Q0) (see [7, Theorem 2.2]). Theorem 5.2. The center of the algebra AW (3, Q0) consists of the scalars. Proof. Let U be in the center of AW (3, Q0). Because of Theorem 2.2 we may consider AW (3, Q0) in its faithful basic representation on Asym. Then U can be uniquely expanded in terms of the basis of AW (3, Q0) in this representation: U = ∑ k,l ak,lD l sym(Z + Z−1)k + ∑ k,l bk,lD l−1 sym(Z + Z−1)Dsym(Z + Z−1)k−1. (5.2) Since U is in the center, we have U(Z + Z−1)− (Z + Z−1)U = 0. (5.3) We will first show that if U given by (5.2) satisfies (5.3), then all coefficients ak,l and bk,l in (5.2) vanish except possibly for coefficients ak,l with l = 0. Indeed, suppose that this is not the case. Then there is a highest value m of k for which ak,l 6= 0 or bk,l 6= 0 for some l ≥ 1. All terms in (5.2) with k > m then will certainly commute with Z + Z−1, so we may assume that the terms in (5.2) with k > m vanish while (5.3) still holds. Let both sides of (5.3) act on the Askey–Wilson polynomial Pj [z]: (U(Z + Z−1)− (Z + Z−1)U)Pj [z] = 0. Expand the left-hand side of the above equation in terms of Askey–Wilson polynomials Pi[z]. Then the highest occurring term will be for i = j + m + 1, so the coefficient of Pj+m+1[z] in this expansion must be zero. This gives∑ l (am,lλ l j+m+1 + bm,lλ l−1 j+m+1λj+m)− ∑ l (am,lλ l j+m + bm,lλ l−1 j+mλj+m−1) = 0. (5.4) We have, writing x := qj+m and u := q−1abcd, λj+m+1 = q−1x−1 + qux, λj+m = x−1 + ux, λj+m−1 = qx−1 + q−1ux. 16 T.H. Koornwinder We can consider the identity (5.4) 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 > 0 for which am,l 6= 0 or bm,l 6= 0. Then, in particular, the coefficients of x−n and xn in the left-hand side of (5.4) must be zero. This gives explicitly: am,nun(1− qn) + q−1bm,nun(1− qn) = 0, am,n(1− q−n) + qbm,n(1− q−n) = 0. Now n > 0, so q±n 6= 1. Also u 6= 0. Hence, am,n + q−1bm,n = 0, am,n + qbm,n = 0. Thus am,n = 0 = bm,n, which is a contradiction. So U in the center will have the form U = ∑ k ak(Z + Z−1)k in the basic representation of AW (3, Q0). We have to show that ak = 0 for k > 0. Suppose not. Then there is a highest value m > 0 of k for which ak 6= 0. Then we have UDsym(1)−Dsym U(1) = 0. Expand the left-hand side of the above equation in terms of Askey–Wilson polynomials Pi[z]. Then the coefficient of Pm[z] will be zero. So amλ0 − amλm = 0. Since λm 6= λ0 if m 6= 0, we conclude that am = 0, a contradiction. � Theorem 5.3. The center Z(H̃) of H̃ consists of the scalars. Proof. Let U ∈ Z(H̃). Then U ∈ ZH̃(T1) = ÃW (3, Q0). So U ∈ Z(ÃW (3, Q0)). Write U as in (5.1). We have to show that U(T1 + 1) and U(T1 + ab) are scalars. This follows from U(T1 + 1) = (1− ab)−1S(U) ∈ Z(S(H̃)), U(T1 + ab) = (ab− 1)−1S−(U) ∈ Z(S−(H̃)). Now use the algebra isomorphisms from Theorems 3.2 and 4.2, and apply Theorem 5.2. � Acknowledgements I thank Jasper Stokman for suggesting me that the spherical subalgebra of the Askey–Wilson DAHA is related to Zhedanov’s algebra. I thank a referee for suggestions which led to inclusion of Remarks 2.1, 2.7 and 4.5. Some of the results presented here were obtained during the work- shop Applications of Macdonald Polynomials, September 9–14, 2007 at the Banff International Research Station (BIRS). I thank the organizers for inviting me. References [1] Berest Y., Etingof P., Ginzburg V., Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337, math.QA/0111005. [2] Cherednik I., Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators, Int. Math. Res. Not. 1992 (1992), no. 9, 171–180. 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Math. 254 (2000), 395–411. [14] Sahi S., Raising and lowering operators for Askey–Wilson polynomials, SIGMA 3 (2007), 002, 11 pages, math.QA/0701134. [15] Zhedanov A.S., “Hidden symmetry” of Askey–Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146–1157. http://arxiv.org/abs/math.QA/0612730 http://arxiv.org/abs/math.QA/0001033 http://arxiv.org/abs/math.RT/0306393 http://arxiv.org/abs/q-alg/9710032 http://arxiv.org/abs/math.QA/0701134 1 Introduction 2 Summary of earlier results 2.1 Askey-Wilson polynomials 2.2 Zhedanov's algebra 2.3 The double affine Hecke algebra of type (C1,C1) 3 The spherical subalgebra 4 The antispherical subalgebra 5 Centralizers and centers References

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

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