Zhedanov s Askey-Wilson algebra, Cherednik s double affine Hecke algebras, and bispectrality. lecture 3: Double affine Hecke algebras.

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1 Zhedanov s Askey-Wilson algebra, Cherednik s double affine Hecke algebras, and bispectrality. Lecture 3: Double affine Hecke algebras in the rank one case University of Amsterdam, T.H.Koornwinder@uva.nl, course of 3 lectures during the International Mathematical Meeting IMM 09 on Harmonic Analysis & Partial Differential Equations, Marrakech, Morocco, April 1 4, 2009 Version of June 28, 2010 Note: This lecture is work in collaboration with Fethi Bouzeffour (Bizerte, Tunisia)

2 Hankel transform Normalized Bessel function: ( 1 4 J α (x) := x 2 ) k ( ) (α + 1) k k! = 0F 1 α + 1 ; 1 4 x 2. k=0 J α (x) = J α ( x), J α (0) = 1, J 1 (x) = cos x, J 1 (x) = sin x 2 2 x. Eigenfunctions: ( d 2 dx 2 + 2α + 1 x ) d J α (λx) = λ 2 J α (λx). dx Hankel transform pair: f (λ) = f (x)j α (λx)x 2α+1 dx, f (x) = α+1 Γ(α + 1) 2 0 f (λ)jα (λx)λ 2α+1 dλ.

3 Non-symmetric Hankel transform Non-symmetric Bessel function: i x E α (x) := J α (x) + 2(α + 1) J α+1(x), so E 1 (x) = e ix. 2 Non-symmetric Hankel transform pair: f (λ) = f (x) E α ( λx) x 2α+1 dx, f (x) = 1 2 2(α+1) Γ(α + 1) 2 Differential-reflection operator: f (λ) Eα (λx) λ 2α+1 dλ. (Yf )(x) := f (x) + (α + 1 f (x) f ( x) 2 ) x (Dunkl operator for root system A 1 ). Eigenfunctions: Y (E α (λ. )) = i λ E α (λ. ).

4 Askey-Wilson polynomials Recall the Askey-Wilson polynomials P n [z] as monic symmetric Laurent polynomials: ( P n [z] = P n [z; a, b, c, d q] = P 1 n 2 (z + z 1 ) ) := (ab, ac, ad; q) ( n q n, q n 1 abcd, az, az 1 ) a n (abcdq n 1 4φ 3 ; q, q. ; q) n ab, ac, ad They satisfy (LP n )[z] = λ n P n [z] with λ n = q n + abcdq n 1 and (Lf )[z] := A[z] f [qz] + A[z 1 ] f [q 1 z] (A[z] + A[z 1 ]) f [z] with A[z] := + (1 + q 1 abcd)f [z] (1 az)(1 bz)(1 cz)(1 dz) (1 z 2 )(1 qz 2 ).

5 Non-symmetric Askey-Wilson polynomials Assumptions q 0, q m 1 (m = 1, 2,...), a, b, c, d 0, abcd q m (m = 0, 1, 2,...), {a, b} {a 1, b 1 } =. In terms of P n [z] = P n [z; a, b, c, d q], Q n [z] := a 1 b 1 z 1 (1 az)(1 bz) P n 1 [z; qa, qb, c, d q] the nonsymmetric Askey-Wilson polynomials are defined by: E n := ab ab 1 (P n Q n ) (n = 1, 2,...), E 0 [z] := 1, E n := (1 qn ab)(1 q n 1 abcd) (1 ab)(1 q 2n 1 abcd) P n ab(1 qn )(1 q n 1 cd) (1 ab)(1 q 2n 1 abcd) Q n (n = 1, 2,...).

6 Eigenfunctions of q-difference-reflection operator Let (Yf )[z] := z( 1 + ab (a + b)z )( (c + d)q (cd + q)z ) q(1 z 2 )(q z 2 ) then + (1 az)(1 b)(1 cz)(1 dz) (1 z 2 )(1 qz 2 f [qz] ) + (1 az)(1 bz)( (c + d)qz (cd + q) ) q(1 z 2 )(1 qz 2 ) + (c z)(d z)( 1 + ab (a + b)z ) (1 z 2 )(q z 2 ) YE n = q n E n (n = 1, 2,...), YE n = q n 1 abcd E n (n = 0, 1, 2,...). f [qz 1 ], f [z] f [z 1 ] These come from I. Cherednik s theory of double affine Hecke algebras associated with root systems, extended by S. Sahi to the type (C l, C l ). Here his case l = 1 is considered.

7 Double affine Hecke algebra of type (C 1, C 1) This is the algebra H generated by Z, Z 1, T 1, T 0 with relations ZZ 1 = 1 = Z 1 Z and (T 1 + ab)(t 1 + 1) = 0, (T 0 + q 1 cd)(t 0 + 1) = 0, (T 1 Z + a)(t 1 Z + b) = 0, (qt 0 Z 1 + c)(qt 0 Z 1 + d) = 0. This algebra acts faithfully on the linear space A of Laurent polynomials: (Zf )[z] := z f [z], (T 1 f )[z] := (a + b)z (1 + ab) (1 az)(1 bz) 1 z 2 f [z] + 1 z 2 f [z 1 ], (T 0 f )[z] := q 1 z((cd + q)z (c + d)q) q z 2 Then Y = T 1 T 0. f [z] (c z)(d z) q z 2 f [qz 1 ].

8 Eigenspaces of T 1 T 1 acting on A has eigenvalues ab and 1. T 1 f = ab f f A sym (symmetric Laurent polynomial). T 1 f = f f [z] = z 1 (1 az)(1 bz)g[z] for some g A sym. Let A be an operator on A. Write f [z] = f 1 [z] + z 1 (1 az)(1 bz)f 2 [z] (f 1, f 2 A sym ). Then we can write (Af )[z] = (A 11 f 1 +A 12 f 2 )[z]+z 1 (1 az)(1 bz)(a 21 f 1 +A 22 f 2 )[z], where the A i j are operators on A sym. So ( ) A11 A f (f 1, f 2 ), A 12. A 21 A 22

9 Rewriting the eigenvalue equation for E n in matrix form (( ) ) ( ) Y11 Y 12 q n P n [z; a, b, c, d q] Y 21 Y 22 a 1 b 1 = 0, P n 1 [z; qa, qb, c, d q] (( ) ) Y11 Y 12 q n 1 abcd Y 21 Y 22 ( ) (1 q n ab)(1 q n 1 abcd) P n [z; a, b, c, d q] (1 q n )(1 q n 1 = 0. cd) P n 1 [z; qa, qb, c, d q] Here Y 11 = ab(1 + q 1 cd) abl a,b,c,d;q, 1 ab Y 22 = ab(1 + q 1 cd) + q 1 L aq,bq,c,d;q, 1 ab where L a,b,c,d;q is the second order q-difference operator L having the p n [z; a, b, c, d q] as eigenfunctions. ( ) Also: Y + q 1 abcdy 1 La,b,c,d;q 0 =. 0 q L aq,bq,c,d;q

10 The off-diagonal operators Y 21 and Y 12 (Y 21 g)[z] = z(c z)(d z) ( g[q 1 z] g[z] ) (1 ab)(1 z 2 )(1 qz 2 ) + z(1 cz)(1 dz) ( g[qz] g[z] ) (1 ab)(1 z 2 )(1 qz 2, ) (Y 12 h)[z] = ab(a z)(b z)(1 az)(1 bz) (1 ab)z(q z 2 )(1 qz 2 ) ( (cd + q)(1 + z 2 ) (1 + q)(c + d)z ) h[z] ab(a z)(b z)(c z)(d z)(aq z)(bq z) q(1 ab)z(1 z 2 )(q z 2 h[q 1 z] ) ab(1 az)(1 bz)(1 cz)(1 dz)(1 aqz)(1 bqz) q(1 ab)z(1 z 2 )(1 qz 2 h[qz]. )

11 An equivalent form for the eigenvalue equations The eigenvalue equations for E n and for E n are equivalent to the four equations L a,b,c,d;q P n [. ; a, b, c, d q] = (q n + abcdq n 1 )P n [. ; a, b, c, d q], L qa,qb,c,d;q P n 1 [. ; qa, qb, c, d q] Y 21 P n [. ; a, b, c, d q] = (q n+1 + abcdq n )P n 1 [. ; qa, qb, c, d q], = (q n 1)(1 cdq n 1 ) P n 1 [. ; qa, qb, c, d q], 1 ab Y 12 P n 1 [. ; qa, qb, c, d q] = ab(q n ab)(1 abcdq n 1 ) 1 ab Note that Y 21 and Y 12 act as shift operators. P n [. ; a, b, c, d q].

12 Askey-Wilson polynomials: orthogonality Askey-Wilson polynomials P n [z] satisfy the orthogonality relation 1 P n [z] P m [z] w[z] dz 4πi C z = h n δ n,m, where (z 2, z 2 ; q) w(z) := (az, az 1, bz, bz 1, cz, cz 1, dz, dz 1, ; q) h 0 = (abcd; q) (q, ab, ac, ad, bc, bd, cd; q), h n := h n h 0 = (q, ab, ac, ad, bc, bd, cd; q) n (abcd; q) 2n (q n 1 abcd; q) n. Here C is the unit circle traversed in positive direction with deformations to separate the sequences of poles converging to zero from the sequences of poles diverging to. For suitable a, b, c, d this can be rewritten as an orthogonality relation for the P n (x) with respect to a positive measure µ supported on [ 1, 1] (or on its union with a finite discrete set).

13 Askey-Wilson polynomials: orthogonality (continued) Let.,. a,b,c,d;q be the Hermitian inner product on A sym such that the P n [. ; a, b, c, d q] are orthogonal in the familiar way: P n [. ; a, b, c, d q], P m [. ; a, b, c, d q] a,b,c,d;q = where h a,b,c,d;q n = (q, ab, ac, ad, bc, bd, cd; q) n (abcd; q) 2n (q n 1 abcd; q) n. (Assume that a, b, c, d are such that.,. a,b,c,d;q and.,. qa,qb,c,d;q are positive definite.) Then h a,b,c,d;q n h qa,qb,c,d;q n 1 = (1 qn )(1 q n 1 cd) (1 q n ab)(1 q n 1 abcd) h a,b,c,d;q n δ n,m, (1 ab)(1 qab)(1 ac)(1 ad)(1 bc)(1 bd) (1 abcd)(1 qabcd).

14 Orthogonality relations: vector-valued case For g 1, h 1, g 2, h 2 symmetric Laurent polynomials define an hermitian inner product (g 1, h 1 ), (g 2, h 2 ) := g 1, g 2 a,b,c,d;q h 1, h 2 qa,qb,c,d;q (1 ab)(1 qab)(1 ac)(1 ad)(1 bc)(1 bd) ab. (1 abcd)(1 qabcd) Then the E n (n Z) in vector-valued form are orthogonal with respect to this inner product. If ab < 0 then the inner product is positive definite. In earlier papers (Sahi, Noumi & Stokman, Macdonald s 2003 book) a biorthogonality was given in the form of a contour integral, and there were no results on positive definiteness of the inner product.

15 Going down in the (q-)askey scheme Askey-Wilson little q-jacobi Jacobi Jackson s 3 rd q-bessel function Bessel function

16 Comment on previous sheet I gave only a very limited selection of possible paths downwards in the (q-)askey scheme. Moreover I moved out of this scheme by taking limits to the non-polynomial orthogonal system of Jackson s q-bessel functions (see Koornwinder & Swarttouw, 1992 for this limit), or to a generalized orthogonal system of classical Bessel functions. The machinery of an algebra given by generators and relations acting on the space of 2-vector-valued polynomials and giving rise to an explicit orthogonal system of eigenfunctions has good limits when going from Askey-Wilson polynomials to little q-jacobi polynomials, from little q-jacobi polynomials to Jacobi polynomials, and from Jacobi polynomials to Bessel functions (arriving at the Dunkl operator as a 2 2 matrix operator acting on 2-vectors of symmetric functions). However, in the limit from little q-jacobi polynomials to Jackson s third q-bessel functions undesired degenerations occur. Work on this is still in progress.

17 Zhedanov s algebra AW (3) generators K 0, K 1, K 2, structure constants B, C 0, C 1, D 0, D 1, relations [K 0, K 1 ] q = K 2, Put for the structure constants: [K 1, K 2 ] q = BK 1 + C 0 K 0 + D 0, [K 2, K 0 ] q = BK 0 + C 1 K 1 + D 1. B := (1 q 1 ) 2 (e 3 + qe 1 ), C 0 := (q q 1 ) 2, C 1 := q 1 (q q 1 ) 2 e 4, D 0 := q 3 (1 q) 2 (1 + q)(e 4 + qe 2 + q 2 ), D 1 := q 3 (1 q) 2 (1 + q)(e 1 e 4 + qe 3 ).

18 Central extension of AW (3) Let the algebra ÃW (3) be generated by K 0, K 1, T 1 such that T 1 commutes with K 0, K 1 and with further relations (T 1 + ab)(t 1 + 1) = 0, (q + q 1 )K 1 K 0 K 1 K 2 1 K 0 K 0 K 2 1 = B K 1 + C 0 K 0 + D 0 + E K 1 (T 1 + ab) + F 0 (T 1 + ab), (q + q 1 )K 0 K 1 K 0 K 2 0 K 1 K 1 K 2 0 = B K 0 + C 1 K 1 + D 1 + E K 0 (T 1 + ab) + F 1 (T 1 + ab), where E := q 2 (1 q) 3 (c + d), F 0 := q 3 (1 q) 3 (1 + q)(cd + q), F 1 := q 3 (1 q) 3 (1 + q)(a + b)cd.

19 Basic representation of ÃW (3) The following element Q commutes with all elements of ÃW (3): Q :=(K 1 K 0 ) 2 (q q 2 )K 0 (K 1 K 0 )K 1 + (q + q 1 )K 2 0 K (q + q 1 )(C 0 K C 1K 2 1 ) + ( B + E(T 1 + ab) )( (q q 1 )K 0 K 1 + K 1 K 0 ) + (q q 1 ) ( D 0 + F 0 (T 1 + ab) ) K 0 + (q q 1 ) ( D 1 + F 1 (T 1 + ab) ) K 1 + G(T 1 + ab), where G can be explicitly specified. ÃW (3) acts on A such that K 0, K 1, T 1 act as Y + q 1 abcdy 1, Z + Z 1, T 1, respectively, in the basic representation of H on A. This action is called the basic representation of ÃW (3) on A. Then Q acts as the constant Q 0 = q 4 (1 q) 2( q 4 (e 4 e 2 ) + q 3 (e 2 1 e 1e 3 2e 2 ) q 2 (e 2 e 4 + 2e 4 + e 2 ) + q(e 2 3 2e 2e 4 e 1 e 3 ) + e 4 (e 1 e 2 ) ).

20 A faithful representation on A Definition ÃW (3, Q 0 ) is the algebra ÃW (3) with additional relation Q = Q 0. Theorem (THK, 2007) ÃW (3, Q 0 ) has the elements K n 0 (K 1K 0 ) i K m 1 T j 1 (m, n = 0, 1, 2,..., i, j = 0, 1) as a linear basis. The polynomial representation of ÃW (3, Q 0) on A is faithful. ÃW (3, Q 0 ) has an injective embedding in H such that K 0 Y + q 1 abcdy 1, K 1 Z + Z 1, T 1 T 1.

21 The spherical subalgebra Definition The spherical subalgebra of H consists of all linear ( operators ) A11 A A 11 on A sym obtained from all linear operators 12 in A 21 A 22 the 2 2 matrix realization of the basic representation of H. Theorem (THK, 2008) The spherical subalgebra is indeed an associative algebra with identity. It coincides with the algebra AW (3, Q 0 ) (defined in lecture 2) in its basic representation.

22 Some literature 1 G. Gasper and M. Rahman, Basic hypergeometric series, 2nd edn., Cambridge University Press, Ya. I. Granovskii, I. M. Lutzenko & A. S. Zhedanov, Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 4 T. H. Koornwinder, The structure relation for Askey-Wilson polynomials, J. Comput. Appl. Math. 207 (2007), T. H. Koornwinder, The relationship between Zhedanov s algebra AW(3) and the double affine Hecke algebra in the rank one case, SIGMA 3 (2007), 063; arxiv:math/ v4. 6 T. H. Koornwinder, Zhedanov s algebra AW(3) and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra, SIGMA 4 (2008), 052; arxiv: v3.

23 Some literature (continued) 7 I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press, M. Noumi and J. V. Stokman, Askey-Wilson polynomials: an affine Hecke algebraic approach, arxiv:math.qa/ S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), S. Sahi, Some properties of Koornwinder polynomials, Contemporary Math. 254, Amer. Math. Soc., 2000, pp L. Vinet & A. Zhedanov, Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math. 211 (2008), A. S. Zhedanov, Hidden symmetry of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991),