ASKEY-WILSON POLYNOMIALS FOR ROOT SYSTEMS OF TYPE BC. Tom H. Koornwinder

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1 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC om H Koornwinder Abstract his paper introduces a family of Askey-Wilson type orthogonal polynomials in n variables associated with a root system of type BC n he family depends, apart from q, on 5 parameters For n = 1 it specializes to the four-parameter family of one-variable Askey-Wilson polynomials For any n it contains Macdonald s two three-parameter families of orthogonal polynomials associated with a root system of type BC n as special cases 1 Introduction In recent years, some families of orthogonal polynomials associated with root systems were introduced he families studied by Heckman & Opdam [6], [4], [5] become Jacobi polynomials for root system BC 1 he families studied by Macdonald (see [11] for root system A n [12] for general root systems) become continuous q-ultraspherical polynomials for root system A 1 continuous q-jacobi polynomials for root system BC 1 (see Askey & Wilson [1, 4]) For all root systems Macdonald s polynomials tend to the Heckman-Opdam polynomials as q tends to 1 his paper introduces a family of Askey-Wilson type polynomials for root system BC n which depends, apart from q, on 5 parameters For n = 1 it specializes to the four-parameter family of Askey-Wilson polynomials For any n it contains Macdonald s two three-parameter families as special cases: for the pair (BC n, B n ) directly for the pair (BC n, C n ) when q is replaced by q 2 Moreover, the weight function integrated over the orthogonality domain was explicitly evaluated by Gustafson [3] as a generalization of Selberg s beta integral he proofs in this paper are very much inspired by Macdonald s proofs in [12], in particular by his proofs in case of root systems E 8, F 4, G 2, where there is no minuscule fundamental weight available he contents of this paper are as follows Section 2 summarizes Macdonald s results he special case BC n of these results is discussed in 3 the further specialization to BC 1 in 4 he long section 5 introduces Askey-Wilson polynomials for root system BC n, shows that these polynomials are eigenfunctions of a certain difference operator establishes the full orthogonality of the polynomials Finally, in 6, special cases open problems are discussed 1991 Mathematics Subject Classification Primary 33D45, 33D80, 17B20 Key words phrases Askey-Wilson polynomials, Macdonald s orthogonal polynomials associated with root systems, root system of type BC 1

2 2 OM H KOORNWINDER Acknowledgements he research presented here was essentially done several years ago, in December 1987, while the author was visiting the IRMA at the University of Abidjan, Côte d Ivoire I thank prof S ouré for his hospitality I thank prof I G Macdonald for sending me his informal preprints at such an early stage he observation that the Askey-Wilson polynomials for root system BC n also include Macdonald s polynomials for the pair (BC n, C n ), was recently made by J F van Diejen I thank G Heckman the referee for useful comments to an earlier version of this paper 2 Summary of Macdonald s results In this section we summarize Macdonald s [12] results on orthogonal polynomials associated with root systems See Humphreys [7] Bourbaki [2, Chap6] for preliminaries on root systems Let V be a finite dimensional real vector space with inner product, Write v := v, v 1/2 for the norm of v V Write v := 2v/ v 2, 0 v V Let R be a not necessarily reduced root system spanning V Let S be a reduced root system in V such that the set of lines {Rα α R} equals {Rα α S} hen the pair (R, S) is called admissible R S have the same Weyl group W Now, for each α R, there is a (unique) u α > 0 such that α := u 1 α α S Assume that R is irreducible It can be arranged, after possible dilation of R S, that u α takes values in {1, 2} or in {1, 3} Let 0 < q < 1 Put q α := q u α Let α t α be a W-invariant function on R, taking values in (0, 1) (for convenience) hen t α only depends on α Put t α := 1 if α V \R Let k α 0 be such that q k α α = t α Let R + be a choice for the set of positive roots in R Let Q := Z-Span(R), Q + := Z + -Span(R + ) Here, throughout the paper, Z + := {0, 1, 2, } Let P := {λ V λ, α Z α R}, P + := {λ V λ, α Z + α R + } be respectively the weight lattice of R the cone of dominant weights Define a partial order on P by λ µ iff λ µ Q + For λ P let e λ be the function on V defined by e λ (x) := e i λ,x, x V Extend this holomorphically to V +iv If f is a function on V then put (wf)(x) := f(w 1 x) for w W, x V Hence we λ = e wλ Let A be the complex linear span of the e λ (λ P) Let A W denote the space of W-invariants of A Put m λ := W λ 1 w W e wλ = µ Wλ e µ, λ P + Here W λ denotes the stabilizer of λ in W he m λ (λ P + ) form a basis of A W Note that m λ (x) = m λ ( x) (λ P +, x V ) If id W then f(x) = f( x) for f A W In particular, we will then have that m λ is real-valued on V

3 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 3 Let R := {α α R} be the root system dual to R Let Q := Z-Span(R ) hen := V/(2πQ ) is a torus Let ẋ be the image in of x V Let dẋ be the normalized Haar measure on For λ P the function ẋ e λ (x) is well-defined on For a, a 1,, a k C put (a; q) := (1 aq j ), (a 1,, a k ; q) := j=0 k (a i ; q) i=1 Define := α R (t 1/2 2α eα ; q α ) (t α t 1/2 2α eα ; q α ), + := α R + (t 1/2 2α eα ; q α ) (t α t 1/2 2α eα ; q α ) hen = + + Define a hermitian inner product on A W by f, g := W 1 f(ẋ) g(ẋ) (x) dẋ Definition 21 For λ P + let P λ A W be characterized by the two conditions (i) P λ = m λ + µ<λ u λ,µ m µ for certain complex coefficients u λ,µ ; (ii) P λ, m µ = 0 if µ < λ heorem 22 P λ, P µ = 0 if λ µ Define ( v f)(x) := f(x i(log q)v), x, v V, for functions f being analytic on a suitable subset of V + iv containing V Hence v e λ = q v,λ e λ, λ P Let σ V be such that σ, α takes just two values 0 1 as α runs through R + (σ is a so-called minuscule fundamental weight for S ) Such σ exists for all S not being of type E 8, F 4 or G 2 In these last three cases we can choose σ such that σ, α takes values 0, 1 2 as α runs through R + Now put Φ σ := σ + +, E σ f := W σ 1 w W w(φ σ σ f), D σ f := W σ 1 w W w(φ σ ( σ f f)), m σ (λ) := W σ 1 q wσ,λ, w W ρ k := 1 2 k α α α R +

4 4 OM H KOORNWINDER heorem 23 D σ maps A W into itself he P λ are eigenfunctions of D σ with eigenvalue q σ,ρ k ( m σ (λ + ρ k ) m σ (ρ k )) If S is not of type E 8, F 4 or G 2 then E σ maps A W into itself, the P λ are also eigenfunctions of E σ with eigenvalue q σ,ρ k m σ (λ + ρ k ) with E σ (1) scalar D σ = E σ E σ (1) 3 he case R = BC n Identify V with R n let ε 1,, ε n be its stard basis Consider in V the root systems R := {±ε j } {±2ε j } {±ε i ± ε j } i<j of type BC n, S B := {±ε j } {±ε i ± ε j } i<j of type B n S C := {±ε j } { 1 2 (±ε i ± ε j )} i<j of type C n hen S B S C are reduced, R, S B S C have the same Weyl groups in the mappings α u 1 α α of R onto S B onto S C, u α take the values 1 2 Note that R = R that the weight lattice P the root lattice Q of R are both given by ake hen P = Q = {m 1 ε m n ε n m 1,, m n Z} R + := {ε j } {2ε j } {ε i ± ε j } i<j P + = {m 1 ε m n ε n m 1 m 2 m n 0, m 1,, m n Z}, Q + = {m 1 (ε 1 ε 2 ) + + m n 1 (ε n 1 ε n ) + m n ε n m 1,, m n Z + } he torus := V/(2πQ ) becomes R n /(2πZ n ) Recall that we have a partial ordering on P such that λ µ iff λ µ Q + For the pair (R, S B ) we have q ±εj = q, q ±2εj = q 2, q ±εi ±ε j = q, there are three different parameters t α, which we write as (Recall that t α = 1 if α / R) hus a := t ±εj, b := t ±2εj, t := t ±εi ±ε j (31) + = ,

5 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 5 where (32) (33) + 1 := n = + 2 := n (b 1/2 e ε j ; q) (a b 1/2 e ε j ; q) (e 2εj ; q 2 ) (b e 2ε j ; q2 ) (e 2ε j ; q) (q 1/2 e ε j, q 1/2 e ε j, a b 1/2 e ε j, b 1/2 e ε j ; q) α=ε i ±ε j ; i<j (e α ; q) (t e α ; q) For the pair (R, S C ) we have q ±εj = q, q ±2εj = q 2, q ±εi ±ε j = q 2, there are three different parameters t α, which we write as a := t ±εj, b := t ±2εj, t := t ±εi ±ε j hus (31) holds with (34) (35) + 1 := n = + 2 := n (b 1/2 e ε j ; q) (a b 1/2 e ε j ; q) (e 2εj ; q 2 ) (b e 2ε j ; q2 ) (e 2ε j ; q 2 ) (a b 1/2 e ε j, q a b 1/2 e ε j, b 1/2 e ε j, q b 1/2 e ε j ; q2 ) α=ε i ±ε j ; i<j (e α ; q 2 ) (t e α ; q 2 ) Since id W in case of root system BC n, m λ will be real-valued we can read for condition (ii) of Definition 21 that P λ (x) m µ (x) (x) dẋ = 0 if µ < λ For the element σ of 2 we can take ε 1 in the case S B ε 1 + ε ε n in the case S C In both cases σ is minuscule So heorem 23 is valid with these choices of σ In particular, in the case S B the polynomial P λ is eigenfunction of D ε1 with eigenvalue (36) n ( ab t 2n j 1 (q λ j 1) + t j 1 (q λ j 1) ) he choice σ := 2ε 1 in the case S C would give values 0, 1 2 for σ, α as α runs through R + It will turn out in 61 that, in case S C, the P λ are not only eigenfunctions of E ε1 + +ε n but also of D 2ε1

6 6 OM H KOORNWINDER 4 he case R = BC 1 For n = 1 the two root systems S B S C coincide the results of 3 specialize as follows We have = R/(2πZ), P = Q = Z, P + = Z +, the partial order on P is the ordinary total order on Z, { e ilx + e ilx, l = 1, 2,, m l (x) = 1, l = 0 + (e 2ix ; q) (x) = (q 1/2 e ix, q 1/2 e ix, a b 1/2 e ix, b 1/2 e ix ; q) (e 2ix ; q 2 ) = (a b 1/2 e ix, q a b 1/2 e ix, b 1/2 e ix, q b 1/2 e ix ; q 2 ) he inner product for W-invariant functions f, g becomes an integral over the period of 2π-periodic even functions, so it can be written as f, g = 1 2π π 0 f(x) g(x) (x) dx Askey-Wilson polynomials p n (y; a, b, c, d q) (n Z + ) are defined, up to a constant factor, as polynomials of degree n in y which satsify the orthogonality relations (41) π 0 (p n p m )(cos x; a, b, c, d q) (e 2ix ; q) (ae ix, be ix, ce ix, de ix ; q) 2 dx = 0, n m See Askey & Wilson [1] Here a, b, c, d are real, or if complex, appear in complex conjugate pairs, a, b, c, d 1, but the pairwise products of a, b, c, d are not 1 When the condition a, b, c, d 1 on the parameters is dropped, finitely many discrete terms have to be added to the orthogonality relation (41) When we compare the expression for + (x) with the Askey-Wilson weight function we see that Macdonald s polynomials for root system BC 1 coincide, up to a constant factor, with Askey-Wilson polynomials p l (cos x; q 1/2, q 1/2, ab 1/2, b 1/2 q) By Askey & Wilson [1, (416), (417), (420)] the continuous q-jacobi polynomials in M Rahman s notation can be expressed in terms of Askey-Wilson polynomials by P (α,β) l (cos x; q) =const p l (cos x; q 1/2, q 1/2, q α+1/2, q β+1/2 q) =const p l (cos x; q α+1/2, q α+3/2, q β+1/2, q β+3/2 q 2 ) hus, if we put a := q α, b := q 2β, then Macdonald s polynomials for root system BC 1 coincide, up to a constant factor, with continuous q-jacobi polynomals P (α+β 1/2,β 1/2) l (cos x; q) his observation was already made by Macdonald [12, 9]

7 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 7 If n = 1 then, with σ := 1, Φ σ (x) = (1 a b1/2 e ix ) (1 + b 1/2 e ix ) 1 e 2ix hus, if we write R l (e ix ) := P l (x) then heorem 23 yields Φ σ ( x) R l (q 1 e ix ) + Φ σ (x) R l (qe ix ) = (abq l + q l ) R l (e ix ) Compare this with Askey & Wilson [1, (57), (58), (59)]: A( x) ( R l (q 1 e ix ) R l (e ix ) ) + A(x) ( R l (qe ix ) R l (e ix ) ) = (1 q l ) (1 q l 1 abcd) R l (e ix ), where R l (e ix ) := const p l (cos x; a, b, c, d q) A(x) := (1 aeix ) (1 be ix ) (1 ce ix ) (1 de ix ) (1 e 2ix ) (1 qe 2ix ) If c = q 1/2, d = q 1/2 (the continuous q-jacobi case) then A(x) = (1 aeix ) (1 be ix ) 1 e 2ix so A(x) + A( x) = 1 ab, A( x) R l (q 1 e ix ) + A(x) R l (qe ix ) = (q l q l ab) R l (e ix ) hus Macdonald s difference equation for P l in case R = BC 1 coincides with the continuous q-jacobi case of the difference equation for Askey-Wilson polynomials 5 Askey-Wilson polynomials for root system BC n We use the notation of 2 3 Let R + 1 := {2ε j},,n, R + 2 := {ε i ± ε j } 1 i<j n, R 1 := R + 1 ( R+ 1 ), R 2 := R + 2 ( R+ 2 ), R + l := R+ 1 R+ 2, R l := R 1 R 2 = R + l ( R+ l ) Let R be the root system of type BC n of 3 hen R l = {α R 2α / R}, a root system of type C n in V with subsystems R 1 of type na 1 R 2 of type D n (he subscript l sts for long ) Let W be the Weyl group of R l It is a semidirect product of the group of permutations of the coordinates the group

8 8 OM H KOORNWINDER of sign changes of the coordinates Let ρ, ρ 1, ρ 2 denote half the sum of the positive roots of R l, R 1, R 2, respectively hen ρ = ρ 1 + ρ 2 ρ 1 = ε 1 + ε ε n, ρ 2 = (n 1)ε 1 + (n 2)ε ε n 1 Let P, P + the partial order be as in 3 Write ε(w) := det(w) (w W) Let A W,ε consist of all f A such that wf = ε(w)f (w W) Write J λ := w W ε(w) e wλ, λ P he J λ+ρ (λ P + ) form a basis of A W,ε In particular, put δ := J ρ = (e 1 2 α e 1 2 α ) = e ρ (1 e α ) α R + l α R + l χ λ := δ 1 J λ+ρ, λ P he χ λ (λ P + ) are in A W form a basis of A W We have χ λ = m λ + a λ,µ m µ, λ P +, µ P + ; µ<λ for certain complex a λ,µ Fix q (0, 1) a, b, c, d, t C Let (51) + := (e α ; q) (ae 1 2 α, be 1 2 α, ce 1 2 α, de 1 2 α ; q) α R + 1 (52) (x) := + (x) + ( x) α R + 2 (e α ; q) (te α ; q) We are now ready to introduce Askey-Wilson polynomials for root system BC n Definition 51 Assume a, b, c, d are real, or if complex, appear in conjugate pairs, that a, b, c, d 1, but the pairwise products of a, b, c, d are not 1 Assume 1 < t < 1 Let := [ π, π] n V For λ P + define P λ A W by the two conditions (i) P λ = m λ + µ P + ; µ<λ u λ,µ m µ for certain coefficients u λ,µ ; (ii) P λ(x) m µ (x) (x) dx = 0 for µ P +, µ < λ We will generalize the case R = BC n of heorem 22 by showing that the P λ are orthogonal on with respect to the weight function he proof will be based on two lemmas, the first one giving the action of a suitable difference operator on the m λ, the second one showing self-adjointness of this operator with respect to on, when acting on A W Let σ := ε 1, similarly as in 3 for the pair (BC n, B n ) Define Φ σ D σ as in 2, with + being given by (51) hus (53) Φ σ := σ + +, (54) D σ f := W σ 1 w W w(φ σ ( σ f f)) = W σ 1 w W(wΦ σ ) ( wσ f f), f A W

9 Lemma 52 with ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 9 (55) a λ,λ = D σ m λ = µ P + ; µ λ a λ,µ m µ n ( q 1 abcd t 2n j 1 (q λ j 1) + t j 1 (q λ j 1) ) Here a, b, c, d, t may be arbitrarily complex Proof It will be convenient to replace a, b, c, d in the expression (51) for + by a, b, q 1 2 c, q 1 2 d, respectively hus (56) + := α R + 1 (e α ; q) (ae 1 2 α, be 1 2 α, q 1 2 ce 1 2 α, q 1 2 de 1 2 α ; q) By substitution of (56) in (53) we obtain Φ σ = (1 aeε 1 ) (1 + be ε 1 ) (1 q 1 2 ce ε 1 ) (1 + q 1 2 de ε 1 ) (1 e 2ε 1 ) (1 qe 2ε 1) Hence where α R + 2 α=ε 1 ±ε l ; l=2,,n (e α ; q) (te α ; q) 1 te α 1 e α = abcd t 2(n 1) (1 a 1 e ε 1 ) (1 + b 1 e ε 1 ) (1 q 1 2 c 1 e ε 1 ) (1 + q 1 2 d 1 e ε 1 ) (1 e 2ε 1 ) (1 q 1 e 2ε 1 ) 1 t 1 e α 1 e α = abcd t 2(n 1) α=ε 1 ±ε l ; l=2,,n α R t σ,α e α 1 e α n [ (1 a σ,ε j e ε j ) (1 + b σ,εj e ε j ) (1 e 2ε j ) (1 q 1 2 c σ,εj e ε j ) (1 + q 1 2 d σ,εj e ε j ) ] (1 q 1 e 2ε j ) (57) Ψ σ := (abcd) σ,ρ 1 t σ,2ρ 2 e ρ+2ρ 1 Φ σ = δ 1 δ 1 q Ψ σ, α R + 2 (1 t σ,α e α ) n [ (1 qe 2ε j ) (1 a σ,εj e ε j )(1 + b σ,εj e ε j ) δ q is the following element of A W : (1 q 1 2 c σ,ε j e ε j ) (1 + q 1 2 d σ,ε j e ε j ) ] (58) n δ q := (q 1 2 e ε j q 1 2 e ε j ) (q 1 2 e ε j q 1 2 e ε j ) =e 2ρ 1 n (1 qe 2ε j ) (1 q 1 e 2ε j )

10 10 OM H KOORNWINDER If E ( 1 2 R 1) R 2 then write E := α E α Expansion of (57) yields (59) Ψ σ = where E 0,,E R+ 1 F R + 2 c E0,,E 4,F e ρ+2ρ 1 2 E 0 E 1 E 4 F, (510) c E0,,E 4,F := ( 1) E 0 + E 1 + E 3 + F q E E E 4 Now we can rewrite (54) as a σ,ρ 1 E 1 b σ,ρ 1 E 2 c σ,ρ 1 E 3 d σ,ρ 1 E 4 t σ,2ρ 2 F D σ f = δ 1 δ 1 q D σ f with (511) Dσ f := W σ 1 w W ε(w) (wψ σ ) ( wσ f f) Consider (511) with f := m λ (λ P + ) substitute (59) hen D σ m λ = W σ 1 W λ 1 c E0,,E 4,F ε(w 1 ) w 1,w 2 W E 0,,E R+ 1 F R + 2 (q w 1σ,w 2 λ 1) e w 1(ρ+2ρ 1 2 E 0 E 1 E 4 F )+w 2 λ Put w 2 = w 1 w hen (512) Dσ m λ = W σ 1 W λ 1 w W c E0,,E 4,F (q σ,wλ 1) E 0,,E R+ 1 F R + 2 J wλ+ρ+2ρ1 2 E 0 E 1 E 4 F Hence D σ m λ A W,ε Now the J-function in (512) is either 0 or ε(w ) δ χ ν, where w W, ν P + so that w (ν + ρ) = wλ + ρ + 2ρ 1 2 E 0 E 1 E 4 F, (513) ν + ρ = (w ) 1 wλ Now + (w ) 1 (3ρ 1 2 E 0 E 1 E 4 ) + (w ) 1 (ρ 2 F ) n (514) (w ) 1 (3ρ 1 2 E 0 E 1 E 4 ) = (w ) 1 k j ε j = n k j ε j 3ρ 1

11 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 11 with k j, k j { 3, 2, 1, 0, 1, 2, 3}, (515) (w ) 1 (ρ 2 F ) = (w ) 1 α R + 2 k α α = α R + 2 k αα ρ 2 with k α, k α = ±1 2, (516) (w ) 1 wλ λ Substitution of (514), (515), (516) in (513) yields (517) ν + ρ λ + 3ρ 1 + ρ 2 = λ + ρ + 2ρ 1 Hence (518) Dσ m λ = ν P + ; ν λ+2ρ 1 b ν J ν+ρ for certain coefficients b ν In order to compute b λ+2ρ1 observe that equality in (517) holds iff equality holds in (514), (515), (516), ie, iff (w ) 1 w W λ (519) E 0 = E 1 = = E 4 = 1 2 (ρ 1 w ρ 1 ), (520) F = ρ 2 w ρ 2 Hence, by (512), (521) b λ+2ρ1 = W σ 1 W λ 1 w,w W;(w ) 1 w W λ ε(w ) c E0,,E 4,F (q σ,wλ 1), where E 0,, E 4, F are determined by (519), (520) It follows from (519), (520) that 2 E 0 + F = ρ w ρ, hence ( 1) E 0 + F = ε(w ) Substitution of (519), (520) into (510) now yields: c E0,,E 4,F = ε(w ) (abcd) 1 2 (1+ (w ) 1 σ,ρ 1 ) t n 1+ (w ) 1 σ,ρ 2 When we substitute this last expression into (521) then we obtain b λ+2ρ1 = W σ 1 W λ 1 (abcd) 1 2 (1+ (w ) 1 σ,ρ 1 ) t n 1+ (w ) 1 σ,ρ 2 (q (w ) 1 σ,λ 1) w,w W;(w ) 1 w W λ = W σ 1 w W(abcd) 1 2 (1+ wσ,ρ 1 ) t n 1+ wσ,ρ 2 (q wσ,λ 1)

12 12 OM H KOORNWINDER Hence (522) b λ+2ρ1 = n (abcd) 1 2 (1+ε) t n 1+ε(n j) (q ελ j 1), ε=±1 which is (55), when we take in account the replacement made for a, b, c, d Next we show that, for f A W, D σ f given by (511) is divisible by δ q In view of (58) this will follow if we can show that, for each w W, (wψ σ ) ( wσ f f) is divisible by the 4n prime factors 1 ± q ±1 2 e ε j By (57), all but the two factors 1 ± q 1 2e wσ are divisors of wψ σ We will show that these two factors are divisors of wσ f f Write f as a Laurent polynomial F(e ε 1,, e ε n ), invariant under the transformations e ε j e ε j If wσ = ε j then wσ f f = F(e ε 1,, qe ε j,, e ε n ) F(e ε 1,, e ε j,, e ε n ) becomes 0 for e ε j = ±q 1 2, hence it is divisible by 1±q 1 2 e ε j A similar argument is valid for wσ = ε j By (518), (523) δ 1 Dσ m λ = c ν m ν ν P + ; ν λ+2ρ 1 for certain coefficients c ν, with c λ+2ρ1 = b λ+2ρ1 given by (522) Also, δ 1 Dσ m λ will still be divisible by δ q By (58), δ q A W with highest term m 2ρ1 Hence D σ m λ = δ 1 δq 1 D σ m λ will be in A W with highest term b λ+2ρ1 m λ We have Hence = α R 1 (eα ; q) (ae 1 2 α, be 1 2 α, ce 1 2 α, de 1 2 α ; q) (eα ; q) (te α ; q) α R 2 (x) = (w )(x) = (w + )(x) (w + )( x), w W Lemma 53 With the assumptions of Definition 51 we have (524) (D σ f)(x) g(x) (x) dx = f(x) (D σ g)(x) (x) dx, f, g A W Proof Since id W, f(x) = f( x) g(x) = g( x) By (54) (53), formula (524) can be equivalently written as (525) ( wσ (w + ))(x) ( ( wσ f)(x) f(x) ) (w + )( x) g( x) dx w W = w W (w + )(x) f(x) ( wσ (w + ))( x) ( ( wσ g)( x) g( x) ) dx Since, f g are W-invariant, formula (525) will be implied by the two identities ( + f)(x i(log q)σ) ( + g)( x) dx = ( + f)(x) ( + g)( x i(log q)σ) dx

13 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 13 w W( σ + )(w 1 x) + ( w 1 x) = w W + (w 1 x) ( σ + )( w 1 x) he second identity is obvious, since id W For the first identity observe that the integral ( + f)(z i(log q), x 2,, x n ) ( + g)( z, x 2,, x n ) dz C over the contour C = [ π, π] [π, π + i log q] [π + i log q, π + i log q] [ π + i log q, π] vanishes by Cauchy s theorem (By the assumptions on a, b, c, d, t there are no singularities inside the contour) Now the result follows, since + f + g are invariant under translations by 2πσ It follows now immediately from Lemmas that: heorem 54 D σ P λ = a λ,λ P λ with a λ,λ given by (55) Now we are ready for the main theorem heorem 55 If λ, µ P +, λ µ, then P λ (x) P µ (x) (x) dx = 0 Proof All integrals m λ (x) m µ (x) (x) dx are continuous in a, b, c, d, t Hence the coefficients u λ,µ in Definition 51 are continuous in a, b, c, d, t his implies that P λ (x) P µ (x) (x) dx is continuous in a, b, c, d, t By heorem 54 Lemma 53, P λ (x) P µ (x) (x) dx = 0 if a λ,λ a µ,µ Fix distinct λ µ it follows from (55) that, for fixed nonzero a, b, c, d, the eigenvalues a λ,λ a µ,µ are distinct as polynomials in t his implies the orthogonality of P λ P µ for a, b, c, d, t in a dense subset of the parameter domain under consideration Hence, by continuity, the theorem follows he method of proof in this last theorem is different from the method used in similar situations by Macdonald [12] While Macdonald leaves the parameters fixed shows that equality of eigenvalues for all q implies (in most cases) equality of weights, the above proof leaves q fixed shows that equality of eigenvalues for all parameter values implies equality of weights

14 14 OM H KOORNWINDER 6 Discussion of results 61 Special cases When we compare (51) with (31), (32) (33), then it is clear that Askey-Wilson polynomials for root system BC n with a, b, c, d, t replaced by q 1 2, q 1 2, ab 1 2, b 1 2, t become Macdonald s polynomials for the pair (BC n, B n ) he operator D σ given by (54) then specializes to the operator for which heorem 23 is valid in case (BC n, B n ), the eigenvalue (55) specializes to the eigenvalue in heorem 23, cf (36) We can also work then with E σ instead of D σ Next, when we compare (51) with (31), (34) (35) then it is clear that our polynomials with a, b, c, d, t, q replaced by ab 1 2,qab 1 2, b 1 2, qb 1 2, t, q 2 become Macdonald s polynomials for the pair (BC n, C n ) he operator D σ given by (54) then becomes the operator D 2ε1 for the pair (BC n, C n ) heorem 23 does not say anything about eigenfunctions of this operator, but heorem 54 implies that Macdonald s polynomials for the pair (BC n, C n ) are eigenfunctions of D 2ε1 his corresponds nicely with the cases E 8, F 4, G 2 of heorem 23, where σ, α takes values 0, 1, 2 as α runs through R + we have to work with D σ instead of E σ It would be interesting to consider if P λ might also be eigenfunction of D σ for other quasi-minuscule σ Comparison of (51) (41) makes it evident that the BC n Askey-Wilson polynomials reduce to the one-variable Askey-Wilson polynomials for n = 1 62 A Selberg-type integral a conjectured quadratic norm Let be given by (52) (51) let the parameters satisfy the inequalities of Definition 51 Gustafson [3, (2)] evaluated the Selberg type integral 1 2π 2π (x) dx = 2 n n! (2π) n 0 0 n (t, t n+j 2 abcd; q) (t j, q, abt j 1, act j 1,, cdt j 1 ; q) On the other h, Macdonald [12, (126)] conjectured an explicit expression for the quadratic norm P λ, P λ for polynomials P λ associated with any admissible pair (R, S) It can be shown that Macdonald s conjecture in case λ = 0 (R, S) = (BC n, B n ) coincides with Gustafson s formula for (a, b, c, d) = (q 1 2, q 1 2, ab 1 2, b 1 2 ) In October 1991, when prof Macdonald was visiting he Netherls, first the author has given a conjectured expression for P λ, P λ / 1, 1, where P λ is an Askey- Wilson polynomial for root system BC n, next Macdonald [13] has rewritten this as a conjectured expression for P λ, P λ On the same occasion, Macdonald [13] has also extended his other conjectures in [12, 12] to the BC n Askey-Wilson case 63 he Askey-Wilson hierarchy for BC n It is very probable that all specializations limit cases of one-variable Askey-Wilson polynomials have their analogues in the case of BC n Someone should certainly write down the orthogonality relations difference operators with explicit eigenvalues for all these specializations In some cases these explicit formulas may be rigorously proved by straightforward limit transition from the general Askey-Wilson case In other cases, the limit transition may only give a formal proof, for a rigorous derivation, the proofs of the present paper will have to be imitated q-racah-type polynomials for root system BC n should also be obtained Here analytic continuation from the BC n Askey-Wilson polynomials will be needed residues, possibly higher dimensional, will have to be taken Similar problems will

15 ASKEY-WILSON POLYNOMIALS FOR ROO SYSEMS OF YPE BC 15 arise when the condition a, b, c, d 1 is dropped in Definition 51 In the corresponding one-variable case discrete terms are then added to the orthogonality relations 64 Quantum group interpretations It is known from work by Koornwinder [9], [10], Koelink [8] Noumi & Mimachi [15], [16] that one-variable Askey- Wilson polynomials have an interpretation on the quantum group SU q (2) Noumi [14] announces an interpretation of Macdonald s polynomials for root system A n 1 as zonal spherical functions on the quantum analogues of the homogeneous spaces GL(n)/SO(n) GL(2n)/Sp(2n) According to Noumi, this was already done for the quantum analogue of SL(3)/SO(3) by Ueno & akebayashi It would be interesting to find quantum group interpretations of Macdonald s polynomials in case of all root systems, also of the BC n Askey-Wilson polynomials considered in the present paper References 1 R Askey J Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem Amer Math Soc (1985), no N Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, R A Gustafson, A generalization of Selberg s beta integral, Bull Amer Math Soc (N S) 22 (1990), G J Heckman, Root systems hypergeometric functions II, Compositio Math 64 (1987), , An elementary approach to the hypergeometric shift operators of Opdam, Invent Math 103 (1991), G J Heckman E M Opdam, Root systems hypergeometric functions I, Compositio Math 64 (1987), J E Humphreys, Introduction to Lie algebras representation theory, Springer, H Koelink, he addition formula for continuous q-legendre polynomials associated spherical elements on the SU(2) quantum group related to Askey-Wilson polynomials, Univ of Leiden, Dept of Math, preprint W-90-11, H Koornwinder, Orthogonal polynomials in connection with quantum groups, Orthogonal polynomials: theory practice (P Nevai, ed), NAO ASI Series C 294, Kluwer, 1990, pp , Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group, CWI Rep AM-R9013, preprint, I G Macdonald, A new class of symmetric functions, Séminaire Lotharingien de Combinatoire, 20e Session (L Cerlienco D Foata, eds), Publication de l Institut de Recherche Mathématique Avancée 372/S-20, Strasbourg, 1988, pp , Orthogonal polynomials associated with root systems, preprint, , Some conjectures for Koornwinder s orthogonal polynomials, informal manuscript, October M Noumi, Macdonald s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, informal summary, March M Noumi K Mimachi, Askey-Wilson polynomials the quantum group SU q (2), Proc Japan Acad Ser A Math Sci 66 (1990), , Rogers q-ultraspherical polynomials on a quantum 2-sphere, Duke Math J 63 (1991), CWI, PO Box 4079, 1009 AB Amsterdam, he Netherls Current address: University of Amsterdam, Faculty of Mathematics Computer Science, Plantage Muidergracht 24, 1018 V Amsterdam, he Netherls address: thk@fwiuvanl