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A q -analogue of the type A Dunkl operator and integral kernel

T.H. Baker? and P.J. Forrester? Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia

arXiv:q-alg/9701039v1 30 Jan 1997

We introduce the q -analogue of the type A Dunkl operators, which are a set of degree–lowering operators on the space of polynomials in n variables. This allows the construction of raising/lowering operators with a simple action on non-symmetric Macdonald polynomials. A bilinear series of non-symmetric Macdonald polynomials is introduced as a q -analogue of the type A Dunkl integral kernel KA (x; y ). The aforementioned operators are used to show that the function satis?es q -analogues of the fundamental properties of KA (x; y ).

1

Introduction

The purpose of this paper is to obtain q -analogues of some fundamental results concerning type A integral kernels KA (x; y ) appearing in the works of Dunkl [5, 6, 7, 4]. The kernel KA allows a multidimensional analogue of the Fourier transform to be constructed, and plays a pivotal role in recent studies of generalized Hermite polynomials [1, 2]. A necessary prerequisite will be to give a suitable de?nition of the q -analogue of the type A Dunkl operator di := ? 1 + ?xi α 1 (1 ? sij ) xi ? xj (1.1)

j =i

where sij acts on functions of x := (x1 , . . . , xn ) by interchanging the variables xi and xj . For the purpose of comparison with later results, we recall some results concerning the kernel KA (x; y ) and the Dunkl operator di . The former is a bilinear series in non-symmetric Jack polynomials, denoted Eη (x), which themselves are eigenfunctions of the Cherednik operators [3] ξi = αxi ? xi xp + (1 ? sip ) + (1 ? sip ) + 1 ? i ?xi p<i xi ? xp x ? xp p>i i

? ?

(1.2)

The Dunkl operator di is related to the Cherednik operator ξi via 1 ? di = sip ? ξi + n ? 1 ? αxi p>i (1.3)

and this will be our starting point in de?ning an suitable q -analogue of di . Following Sahi [17], for a node s = (i, j ) in a composition η := (η1 , η2 , . . . , ηn ) ∈ INn , de?ne the arm length a(s), arm colength a′ (s), leg length l(s) and leg colength l′ (s) by a(s) = ηi ? j a (s) = j ? 1

? ?

l(s) = #{k > i|j ≤ ηk ≤ ηi } + #{k < i|j ≤ ηk + 1 ≤ ηi } l′ (s) = #{k > i|ηk > ηi } + #{k < i|ηk ≥ ηi } (1.4)

′

email: tbaker@maths.mu.oz.au; supported by the ARC email: matpjf@maths.mu.oz.au; supported by the ARC

1

Using these, de?ne constants dη :=

s∈η

(α(a(s) + 1) + l(s) + 1) (α(a′ (s) + 1) + n ? l′ (s))

s∈η

d′ η :=

s∈η

(α(a(s) + 1) + l(s)) (1.5)

eη :=

With these constants, the type A kernel is de?ned as [2] KA (x; y ) =

η

α|η|

dη Eη (x) Eη (y ) d′ η eη

(1.6)

Set si := si,i+1 for 1 ≤ i ≤ n ? 1. The following raising/lowering operators Φ := xn sn?1 · · · s2 s1 = sn?1 · · · si xi si?1 · · · s1 Φ := d1 s1 s2 · · · sn?1 = s1 s2 · · · si?1 di si si+1 · · · sn?1 have a very simple action on the non-symmetric Jack polynomials Eη (x) [10, 2], Φ Eη = EΦη 1 d′ η Φ Eη = EΦη α d′

Φη

(1.7) (1.8)

(1.9) (1.10)

where Φη := (η2 , η3 , . . . , ηn , η1 + 1) and Φη := (ηn ? 1, η1 , η2 , . . . , ηn?1 ). The fundamental properties of the kernel KA (x; y ) are given by the following result [2, Theorem 3.8] Theorem 1.1 The function KA (x; y ) possesses the following properties (a) (b) (c) si di

(y )

KA (x; y ) = si

(x)

KA (x; y )

Φ(y) KA (x; y ) = Φ(x) KA (x; y )

(y )

KA (x; y ) = xi KA (x; y )

The above kernel has a symmetric counterpart 0 F0 (x; y ) which itself is expressed in terms of (α) the symmetric Jack polynomials Pλ (x) [15]. The symmetric Jack polynomials can be expressed in terms of their non-symmetric siblings Eη (x) by [17]

(α) Pκ (x) = d′ κ η

1 Eη (x) d′ η

(1.11)

where the sum is over distinct permutations η of the partition κ. They can also be obtained by symmetrization [2] n! eη (α) (1.12) Pη+ (x) Sym Eη (x) = (α) n dη Pλ (1 ) where Sym denotes the operation of symmetrization of the variables x1 , . . . , xn and η + denotes the (unique) partition associated with η , obtained by permuting its entries. It was shown in [2] that the symmetric kernel

0 F0 (x; y ) := κ

α|κ| Pκ (x) Pκ (y ) (α) d′ κ Pκ (1n ) 2

(α)

(α)

can be obtained from the non-symmetric kernel KA (x; y ) via symmetrization: Sym(x) KA (x; y ) = n! 0 F0 (x; y ). (1.13)

In this work, we shall be concerned with providing q -analogues of the above results. After introducing preliminary results and notations dealing with non-symmetric Macdonald polynomials Eη (x; q, t) and (type A) a?ne Hecke algebras, we proceed to de?ne an analogue of the Dunkl operator (1.1) and show that they form a mutually commuting set of degree-lowering operators. We then construct the analogue of Knop and Sahi’s raising operator Φ given by (1.7), as well as its lowering counterpart Φ, demonstrating their simple action on Eη (x; q, t). Finally, we construct a q -analogue of the kernel KA (x; y ) and derive the corresponding version of Theorem 1.1. Symmetrization of this kernel is then shown to recover the well-known symmetric version 0 F0 (x; y ; q, t) [13, 12].

2

Preliminaries

We begin by presenting the standard realization of the a?ne Hecke algebra on the space of polynomials in n variables (see e.g. [9, 11]). Let τi be the q -shift operator in the variable xi , so that (τi f )(x1 , . . . , xi , . . . , xn ) = f (x1 , . . . , qxi , . . . , xn ). The Demazure-Lustig operators are de?ned by Ti = t + and T0 txi ? xi+1 (si ? 1) xi ? xi+1 qtxn ? x1 (s0 ? 1) = t+ qxn ? x1 i = 1, . . . , n ? 1 (2.1) (2.2)

?1 . For future reference, we note the following action of T , 1 ≤ i ≤ n ? 1 on where s0 := s1n τ1 τn i b a the monomial xi xi+1

b Ti xa i xi+1 =

? a?1 b+1 b+1 a?1 b a ? ? (1 ? t)xi xi+1 + · · · + (1 ? t)xi xi+1 + xi xi+1

a txa i xi+1 ? ? (t ? 1)xa xb + · · · + (t ? 1)xb?1 xa+1 + txb xa i i+1 i i+1 i i+1

a>b a=b a<b

(2.3)

In addition to the operators Ti , de?ne

ω := sn?1 · · · s2 s1 τ1 = sn?1 · · · si τi si?1 · · · s1 . The a?ne Hecke algebra is then generated by elements Ti , 0 ≤ i ≤ n ? 1 and ω , satisfying the relations (Ti ? t) (Ti + 1) = 0 Ti Ti+1 Ti = Ti+1 Ti Ti+1 Ti Tj = Tj Ti |i ? j | ≥ 2 ω Ti = Ti?1 ω From the quadratic relation (2.4), we have the identity Ti?1 = t?1 ? 1 + t?1 Ti . (2.8) (2.4) (2.5) (2.6) (2.7)

3

Useful relations between the operators ω , Ti and xi , 1 ≤ i ≤ n ? 1, include Ti?1 xi+1 = t?1 xi Ti Ti xi = txi+1 Ti?1 ω xi = qxn ω We can de?ne operators? [9]

?1 1 Yi = t?n+i Ti · · · Tn?1 ω T1 · · · Ti? ?1 ,

Ti?1 xi = xi+1 Ti?1 + (t?1 ? 1)xi Ti xi+1 = xi Ti + (t ? 1)xi+1 ω xi+1 = xi ω 1≤i≤n?1

(2.9) (2.10) (2.11)

1≤i≤n

(2.12)

which commute amongst themselves: [Yi , Yj ] = 0, 1 ≤ i, j ≤ n. They also possess the following relations with the operators Ti Ti Yi+1 Ti = t Yi [Ti , Yj ] = 0, j = i, i + 1 (2.13)

while the following relations with xn will be needed in Section 4

?1 1 Yi xn = xn Yi + t?n+1 (1 ? t)xn Ti · · · Tn?1 ωT1 · · · Ti? ?1

1≤i≤n?1

(2.14) (2.15)

Yn xn = qt

?n+1

xn ωT1 · · · Tn?1

These identities follow from a direct calculation involving (2.11), (2.10) and (2.8). The fact that the operators Yi mutually commute implies that they possess a set of simultaneous eigenfunctions, the non-symmetric Macdonald polynomials. Speci?cally, let ? denote the partial order on compositions η de?ned for η = ν by ν?η i? ν + < η+ or in the case ν + = η + ν<η

where < is the usual dominance order for n-tuples, i.e. ν < η i? p i=1 (ηi ? νi ) ≥ 0, for all 1 ≤ p ≤ n. Then the non-symmetric Macdonald polynomials Eη (x; q, t) can be de?ned by the conditions Eη (x; q, t) = xη +

ν ?η η ?i

bην xν 1≤i≤n (2.16)

Yi Eη (x; q, t) = t Eη (x; q, t) where

η ?i = αηi ? #{k < i | ηk ≥ ηi } ? #{k > i | ηk > ηi } with α a parameter such that tα = q . De?ne the analogue of the constants (1.5) by [16] dη (q, t) :=

s∈η

1 ? q a(s)+1 tl(s)+1 1?q

s∈η a′ (s)+1 n?l′ (s)

d′ η (q, t) :=

s∈η

1 ? q a(s)+1 tl(s) (2.17)

eη (q, t) :=

t

Certain properties of these coe?cients follow immediately from [17, Lemmas 4.1, 4.2] Lemma 2.1 We have eΦη (q, t) dΦη (q, t) η1 = = 1 ? qtn+? , dη (q, t) eη (q, t) d′ Φη (q, t) η1 = 1 ? qtn?1+? , d′ ( q, t ) η esi η (q, t) = eη (q, t), for ηi > ηi+1

1 ? tδi,η +1 dsi η (q, t) = , dη (q, t) 1 ? tδi,η

?

d′ 1 ? tδi,η si η (q, t) = d′ 1 ? tδi,η ?1 η (q, t)

The normalization is chosen such that as q → 1, (1 ? Yi )/(1 ? q ) → ξi /α with ξi given by (1.2)

4

3

q -Dunkl operators

? ? ? ?

We introduce the q -Dunkl operators Di for 1 ≤ i ≤ n according to

n ?1

Di := where for i < j ,

1 x? i

?1 ? t

n?1 ?

1 + (t

? 1)

j =i+1

t

j ?i

?1 ? ? Tij Yi

(3.1)

?1 1 ?1 ?1 ?1 ?1 Tij := Ti?1 Ti? +1 · · · Tj ?2 Tj ?1 Tj ?2 · · · Ti

=

1 ?1 ?1 ?1 ?1 1 Tj? Ti+1 · · · Tj? ?1 Tj ?2 · · · Ti+1 Ti ?1

?1 Note that as q → 1 in (3.1), since Tij → sij and (1 ? Yi )/(1 ? q ) → ξi /α, we recover the type A Dunkl operators due to the relation (1.3):

q →1

lim

Di = di 1?q

The relations between the operators Di and the elements Ti , ω of the a?ne Hecke algebra are given by the following two lemmas, Lemma 3.1 Ti Di+1 = t Di Ti?1 , [Ti , Dj ] = 0 Ti Di = Di+1 Ti + (t ? 1)Di 1≤i≤n?1 j = i, i + 1 (3.2) (3.3)

Proof. First note that the second relation in (3.2) follows from the ?rst relation by multiplying the latter on the left by Ti?1 , on the right by Ti , and then using (2.8). From (2.8) it follows that Ti?1 obeys the quadratic relation Ti?2 + (1 ? t?1 )Ti?1 ? t?1 = 0. (3.4)

Multiply the relation t?1 Yi+1 = Ti?1 Yi Ti?1 (which follows directly from (2.13)) on the left by Ti?1 and apply (3.4) to give Ti?1 Yi+1 = (t?1 ? 1) Yi+1 + Yi Ti?1 . (3.5)

?1 ?1 1 (which follows from the ?rst From the de?nition (3.1) and the relations Ti x? i+1 = txi Ti equation in (2.10)), (2.13) and (3.5), the ?rst relation in (3.2) can be deduced. Turning to (3.3), note that a convenient representation of Dj for j < n in terms of 1 n?1 Dn := x? Yn ) n (1 ? t

(3.6) (3.7)

is Dj = t?n+j Tj · · · Tn?1 Dn Tn?1 · · · Tj . Thus for i < j ? 1 it follows that [Ti , Dj ] = 0. For j ? 1 ≤ i ≤ n ? 1 we have Ti Dj = t?n+j Tj · · · Tn?1 Ti?1 Dn Tn?1 · · · Tj = t?n+j Tj · · · Tn?1 Dn Ti?1 , Tn?1 · · · Tj = Dj Ti , where the ?rst equality follows from (2.5), the second from the already established commutativity of Ti , Dj for i < j ? 1, and the ?nal equality follows by further use of (2.4). 2 5

Lemma 3.2 We have ω Di+1 = Di ω qω D1 = Dn ω Proof. We ?rst prove (3.8) in the special case i = n ? 1. To this end, note that for i ≥ 2,

?1 ?1 ?1 ω Yi = Yi?1 ω + (1 ? t)Ti?1 · · · Tn?2 Tn ?1 Tn?2 · · · Ti?1 Yi?1 ω

1 ≤i ≤n?1

(3.8) (3.9)

which follows from using (2.7) to shift the operator ω to the right. In particular

?1 ω Yn = Yn?1 ω + (1 ? t)Tn ?1 Yn?1 ω.

(3.10)

The use of (2.11) and (3.10) and the explicit expression for Dn given by (3.6) allows one to show that ω Dn = Dn?1 ω . For the cases i < n ? 1 the result follows from the case i = n ? 1 and the representation of Di in terms of Dn given by (3.7) since ω Di+1 = t?n+i+1 ωTi+1 · · · Tn?1 Dn Tn?1 · · · Ti+1 = t?n+i+1 Ti · · · Tn?2 Dn?1 Tn?2 · · · Ti ω = t?n+i+1 Ti · · · Tn?2 t?1 Tn?1 Dn Tn?1 Tn?2 · · · Ti ω = Di ω To prove (3.9), ?rst note that repeated use of (3.4) yields that, for i < j

?1 1 ?1 ?1 1 ?1 Iij := Ti?1 Ti? · · · Ti? +1 · · · Tj Tj +1 Ti j +1

= It then follows that

ti?j ?1 + (t?1 ? 1)

p=i+1

?1 tp?j ?1 Tip

(3.11)

?1 ?1 n?1 ?1 Yn ω = ω T1 · · · Tn ω I1 ?1 ω = t ,n?1 Y1

and so from this and (3.11), we have

1 1 2n?2 ?1 Dn ω = x? I1,n?1 Y1 1 ? t n ? 1 Y n ω = x? n n ω 1?t 1? 1 ? tn?1 ?1 + (t?1 ? 1) = qω x? 1

?

?

n p=2

?1 ? Y1 ? = qω D1 tp?1 T1 p

?

?

2

Remarks. 1. The ?nal relations between the operators Di , 1 ≤ i ≤ n and the generators of the a?ne Hecke algebra are the ones involving the generator T0 . These takes the form

?1 T0 D1 = q ?1 tDn T0 ,

T0 Dn = qD1 T0 + (t ? 1)Dn 2≤i≤n?1

[T0 , Di ] = 0, 2. It follows from (3.11) that

which follow immediately using the fact that T0 = ω T1 ω ?1 along with Lemmas 3.1 and 3.2.

1 ?1 Di = x? 1 ? t2n?i?1 Ii,n i ?1 Y i 1 ?1 ?1 ?1 = x? 1 ? tn?1 Ti?1 · · · Tn ?1 ω T1 · · · Ti?1 i

(3.12)

providing an alternative de?nition of the q -Dunkl operators. Another set of relations which shall be needed later on is an analogue of [1, Lemma 3.1] 6

Lemma 3.3 We have [Di , Yj ] = ti?j (1 ? t)Yj Tij Dj tj ?i (1 ? t)Yi Tji Di

n

i<j i>j

i?1

(3.13) tp?i Tip Di

p=1

Di Yi ? qYi Di = (t ? 1)

p=i+1

t?p+i Yp Tip Dp + q (t ? 1) Yi

(3.14)

Proof. We start with (3.13) when i < j . In this case, we can use Lemmas 3.1 and 3.2 to shu?e the Di to the right to get

?1 1 ?1 ?1 ?1 Di Yj = t?n+j Tj · · · Tn?1 ω T1 · · · Ti? ?1 Di+1 Ti Ti+1 · · · Tj ?1 ?1 1 ?1 ?1 = Yj Di + t?n+j (t?1 ? 1) Tj · · · Tn?1 ω T1 · · · Ti? ?1 Di+1 Ti+1 · · · Tj ?1

where we have used the fact that Di+1 Ti?1 = Ti?1 Di + (t?1 ? 1)Di+1 (which follows directly from (3.2) ). Using Lemma 3.1 on the second of these terms to move Di+1 to the right results in the term ti?j (1 ? t)Yj Tij Dj as required. The proof in the case i > j is somewhat similar. To prove (3.14) we must also consider 2 cases: 1 ≤ i < n and i = n. For the case 1 ≤ i < n, we use the identity Di Ti = Ti Di+1 + (t ? 1)Di (which can be obtained from Lemma 3.1 ) to move the operator Di to the right in the expression

?1 1 Di Yi = t?n+i Di Ti · · · Tn?1 ω T1 · · · Ti? ?1

and thus obtain

n

Di Yi = (t ? 1)

p=i+1

t?p+i Yp Tip Dp + qt?i+1 Yi Ti?1 · · · T1 T1 · · · Ti?1 Di

The second term in the above expression can be simpli?ed using a result similar to (3.11), namely that for i < j

j

Tj Tj ?1 · · · Ti Ti · · · Tj ?1 Tj = t

j ?i+1

+ (t ? 1)

p =i

tp?i Tp,j +1 2

and the stated result follows. The case i = n is derived similarly.

In the rest of this section we shall show that the q -Dunkl operators Di commute amongst themselves. We do this by showing that all the Di commute with Dn (recall that Dn has the simplest form amongst all the q -Dunkl operators) from which the general result follows swiftly. First note that for i < n

1 ?1 n?i?1 1 ?1 Yi x? (t ? 1)x? n = xn Y i + t i Tin Yi 1 which follows from pulling x? n to the left using 1 ?1 ?1 Ti x? i+1 = txi Ti 1 ?1 Ti x? i = xi+1 Ti + (t ? 1)xi

(3.15)

(3.16)

(which themselves follow from (2.9) and (2.10) ). Using (3.15) and (3.16), a series of manipulations yields the relation 1 2(n?i)?1 ?1 ?1 Yn Tin x? xi Tin Yi n =t

7

Using this and Lemma 3.3 we can rewrite the commutator [Di , Yn ] in the form

1 ?1 n?1 [Di , Yn ] = tn?i (t?1 ? 1) x? ) Yn i Tin Yi (1 ? t

(3.17)

Another result we need is

?1 ?1 Tin , xn ?1 ?1 1 ?1 1 ?1 ? 1) x? = (x? n Ii,n?1 i ? xn ) Tin + (t n?1

+ (t?1 ? 1)

p=i+1

?1 ?1 ?1 ?1 1 ?1 x? p Ti · · · Tp?2 Tp · · · Tn?1 · · · Ti

(3.18)

which can be derived by repeated use of

1 ?1 ?1 1 Ti?1 x? + (t?1 ? 1)x? i+1 = xi Ti i+1

(which itself is derived from (3.16) ). This is used in the derivation of the ?nal necessary ingredient Lemma 3.4 We have for i < n

1 ?1 ?1 1 Di , x? = t2n?i?1 (t?1 ? 1) x? n i xn Tin Yi

Proof.

1 n?1 A Y ) where Write Di = x? i i i (1 ? t n

Ai := 1 + (t?1 ? 1)

j =i+1

?1 tj ?i Tij

Then we have

1 Di , x? n

=

1 n?1 1 x? Ai Yi ), x? n i (1 ? t 1 1 Ai , x? Yi + Ai Yi , x? n n

1 = ?tn?1 x? i

(3.19) (3.20)

?

Note that

?1 ?1 1 Ai , x? = (t?1 ? 1)tn?i Tin , xn n

while from (3.15) we have

1 Ai Yi , x? = (1 ? n

1 ?1 ?1 However, for i < j application of the relation Ti?1 x? i = t xi+1 Ti tells us that ?1 ?1 1 ?1 ?1 Tij xi = t?j +i x? j Ti · · · Tj ?2 Tj ?1 Tj ?2 · · · Ti

1 ?1 t?1 )tn?i ?x? i Tin

?

n

+ (t?1 ? 1)

j =i+1

?1 ?1 ?1 ? tj ?i Tij xi Tin Yi

(3.21)

so that

?1 ?1 ?1 Tij xi Tin =

? ?1 ?1 1 ?1 ?1 ?1 ? t ? j + i x? j Ti · · · Tj ?2 Tj · · · Tn?1 · · · Ti ? t ? n + i x? 1 I ? 1 n i,n?2

j<n j=n

Substituting this into (3.21) and hence into (3.19) along with (3.20) (after using (3.18) ) yields the result. 2 Proposition 3.5 We have [Di , Dj ] = 0 Proof. Consider ?rst the case j = n. In this case [Di , Dn ] = =

1 Di , x? 1 ? t n?1 Y n n 1 Di , x? n 1 1 ? t n ? 1 Y n ? t n ? 1 x? n [Di , Yn ] = 0

1 ≤ i, j ≤ n

thanks to (3.17) and Lemma 3.4. The general result now follows using the representation (3.7) for Dj in terms of Dn . 2 8

4

Raising/lowering operators

The q -analogue of the raising operator Φ (recall (1.7)) introduced by Knop and Sahi [10] is de?ned as ?1 ?1 ?1 1 ?1 ?n+i Φq := xn Tn Tn?1 · · · Ti xi Ti? (4.1) ?1 · · · T2 T1 = t ?1 · · · T1 This operator enjoys the following properties Proposition 4.1 (a) (b) Proof. Yj Φq = Φq Yj +1 Yn Φ q = q Φ q Y1 1≤j ≤n?1

To prove (a), ?rst note from (2.14) that

?1 ?1 Yj Φq = Yj xn Tn ?1 · · · T1 = B1 + B2

where

?1 ?1 B1 = xn Yj Tn ?1 · · · T1 1 ?1 ?1 = Φq Yj +1 + (1 ? t?1 )xn Tn?1 · · · Tj? +1 Tj ?1 · · · T1 Yj +1

and

?1 1 ?1 ?1 B2 = t?n+j (1 ? t)xn Tj · · · Tn?2 ω T1 · · · Tj? ?1 Tn?1 · · · T1 1 ?1 ?1 ?1 = t?n+j (1 ? t)xn Tj? ?1 · · · T1 ω T1 · · · Tj

A careful inspection of the second term occurring in B1 shows that it cancels with B2 , whence the result. Similar considerations follow for (b), with the aid of (2.15). 2 The analogue of (1.9) is given by the following Corollary 4.2 The operator Φq acts on non-symmetric Macdonald polynomials in the following manner Φq Eη (x; q, t) = t?#{ηi ≤η1 } EΦη (x; q, t) where Φη := (η2 , η3 , . . . , ηn , η1 + 1). Proof. From the previous Proposition, it is clear that Φq Eη (x; q, t) is a contant multiple of EΦη (x; q, t) as they are both eigenfunctions of the operators Yi with the same set of eigenvalues. The multiple is deduced by means of examining the coe?cient of the leading term xΦη in the expansion of Φq Eη (x; q, t) with the aid of (2.8) and (2.3). 2 The de?nition of the lowering operator analogous to (1.8) makes use of the q -Dunkl operator introduced in Section 3:

?1 Φq = T1 T2 · · · Tn?1 Dn = tn?i T1 · · · Ti?1 Di Ti?1 · · · Tn ?1

(4.2)

This operator acts as a shift operator for the operators Yi . Proposition 4.3 (a) (b) Yj Φ q = Φ q Y j ?1 Y 1 Φ q = q ?1 Φ q Y n 9 2≤j≤n

Proof.

We begin with (a). Note that for j > 2 we have

1 Tj?1 T1 · · · Tn?1 = T1 · · · Tn?1 Tj? ?1 .

(4.3)

Thus

?1 1 Yj Φq = t?n+j Tj · · · Tn?1 ω T1 · · · Tj? ?1 (T1 · · · Tn ) Dn ?1 ?1 ?1 ?1 · · · Tn = tj ?2 D1 Tj · · · Tn?1 T1 ?2 ω T1 · · · Tj ?2

(4.4)

where we have used (4.3) along with Lemmas 3.1, 3.2 and (2.7) to make the necessary manipulations to get it into the above form. However we can rewrite the term in the parenthesis occurring in (4.4) as

?1 ?1 ?1 ?1 Tj · · · Tn?1 T1 · · · Tn ?2 = T1 · · · Tn?1 Tj ?1 · · · Tn?1

which, when substituted back into (4.4), yields the requisite result. The proof of (b) is almost immediate from (3.9). 2 As a consequence, we have the analogue of (1.10) Corollary 4.4 The action of Φq on non-symmetric Macdonald polynomials is given by Φq Eη (x; q, t) = t#{ηi <ηn } where Φη := (ηn ? 1, η1 , η2 , . . . , ηn?1 ). Proof. From Proposition 4.3 we have that Φq Eη (x; q, t) is a multiple of EΦη (x; q, t). An d′ η (q, t) E (x; q, t) ′ d (q, t) Φη

Φη

examination of the leading term xΦη in the expansion of Φq Eη (x; q, t) using (2.3) and the explicit form (3.6) for Dn tells us that

ηn ) EΦη (x; q, t). Φq Eη (x; q, t) = t#{ηi <ηn } (1 ? tn?1+?

However from Lemma 2.1 we know that d′ η (q, t) ηn = 1 ? tn?1+? ′ d (q, t)

Φη

whence the result.

2

5

Kernels

dη (q, t) ′ dη (q, t)eη (q, t)

We introduce the q -analogue of the Dunkl kernel KA (x; y ) by KA (x; y ; q, t) =

η

Eη (x; q, t)Eη (y ; q ?1 , t?1 )

(5.1)

This function reduces to KA (x; y ) as q → 1 (although KA (x; y ; q, t) = KA (y ; x; q, t) in general) and satis?es generalizations of Theorem 1.1 and (1.13). To establish these generalizations requires properties of the operators Di , Φq and Φq obtained above, as well as some additional

10

properties of the operators Ti (1 ≤ i ≤ n ? 1). The ?rst such property required relates to the action of the operators Ti±1 on the non-symmetric Macdonald polynomials [16]

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Ti Eη =

t?1 1 ? t?δiη t Eη t?1 1 ? t?δiη t?1 ? 1 1 ? tδiη ? t 1 Eη t?1 ? 1 1 ? tδiη

Eη + t Esi η

ηi < ηi+1 (5.2)

ηi = ηi+1 (1 ? tδiη +1 )(1 ? tδiη ?1 ) Eη + Esi η ηi > ηi+1 (1 ? tδiη )2 Eη + Esi η Eη + t?1 ηi < ηi+1 ηi = ηi+1 (1 ? tδiη +1 )(1 ? tδiη ?1 ) Esi η ηi > ηi+1 (1 ? tδiη )2

Ti?1 Eη =

(5.3)

where δiη := η ?i ? η ?i+1 . Now de?ne the involution as acting on operators or functions by sending q → q ?1 , t → t?1 . The following lemma is the analogue of [2, Lemma 3.7] Lemma 5.1 Let F (x, y ) =

η

Aη Eη (x; q, t)Eη (y ; q ?1 , t?1 ). Then (Ti±1 )(x) F (x, y ) = Ti?1 (y) F (x, y ) (5.4)

if and only if the coe?cients Aη satisfy

? (1 ? tδiη )2 ? ? ? ? (1 ? tδiη +1 )(1 ? tδiη ?1 ) Aη = ? ? (1 ? tδiη +1 )(1 ? tδiη ?1 ) ? ? Aη δ 2

ηi > ηi+1 (5.5) ηi < ηi+1

Asi η

(1 ? t

iη

)

Moreover, these two conditions on Aη are equivalent.

Proof. Equation (5.4) consists of two separate equations; only Ti F = Ti?1 F will be established as the other case follows from (2.8). The proof is similar to that given for [2, Lemma 3.7]. (x) Split the sum in Ti F (x, y ) according to whether ηi < ηi+1 , ηi = ηi+1 or ηi > ηi+1 . Apply (5.2) and collect coe?cients of Eη (x; q, t). Also, to work out the action of Ti?1 (y) on Eη (y ; q ?1 , t?1 ) (and hence on F (x, y )), set t → t?1 in (5.3). The two sides of (5.4) are equal if and only if (5.5) holds. 2 With this result at our disposal, the q -analogue of Theorem 1.1 can now be given. Theorem 5.2 The function Kq (x; y ) possesses the following properties: (a) (b) (c) Proof. (a) From Lemma 2.1, the constants Aη = dη (q, t) satisfy the conditions of Lemma 5.1, d′ ( q, t)eη (q, t) η (Ti±1 )(x) KA (x; y ; q, t) = Ti?1

x) Φ( q KA (x; y ; q, t) = Φq (x) (y )

(x)

(y )

KA (x; y ; q, t)

(y )

KA (x; y ; q, t)

Di

KA (x; y ; q, t) = yi KA (x; y ; q, t)

11

hence the result. (b) From Lemma 2.1 and Corollories 4.2, 4.4 we have

x) Φ( q KA (x; y ; q, t) = η ′ dη (q, t) #{ηi <ηn } dη (q, t) ?1 ?1 t ′ (q, t) EΦη (x; q, t)Eη (y ; q , t ) d′ ( q, t ) e ( q, t ) d η η Φη

=

ν

t#{νi ≤ν1 }

(y )

dΦν (q, t) ′ dν (q, t)eΦν (q, t)

Eν (x; q, t)EΦν (y ; q ?1 , t?1 )

= Φq (c) From (4.1) we have

KA (x; y ; q, t)

?1 xi = tn?i Ti?1 · · · Tn ?1 Φq T1 · · · Ti?1 ?1 = t?n+i Ti?1 · · · Tn ?1 Φq T1 · · · Ti?1

where the second form follows from applying the involution

to the ?rst form. Also, from (4.2)

1 ?1 Di = t?n+i Ti? ?1 · · · T1 Φq Tn?1 · · · Ti

The result now follows from these two expressions for xi , Di by means of (a), (b) and the fact that operators acting on di?erent sets of variables commute. 2 It remains to present the analogue of (1.13). For the q -analogue of Sym, Macdonald [14] has introduced the operator U + := Tw (5.6)

w ∈Sn

where w = si1 . . . sip (1 ≤ i1 , . . . , ip ≤ n ? 1) is the reduced decomposition in terms of elementary transpositions of each element of Sn and Tw = Ti1 . . . Tip (5.7)

(the operators Ti used by Macdonald satisfy (Ti ? t)(Ti + t?1 ) = 0 as distinct from (2.4); consequently in [14] U + is de?ned with Tw multiplied by t?(w) , where ?(w) is the length of the permutation w, i.e. the number of elementary transpositions in its reduced decomposition). As noted in [14], use of (2.4) and (2.5) shows that Ti U + = tU + which from the de?nition (2.1) implies that U + f is symmetric in x1 , . . . , xn . In particular, for some proportionality constant aη (q, t), we must have (5.8) U + Eη (x; q, t) = aη (q, t)Pη+ (x; q, t) where Pη+ denotes the symmetric Macdonald polynomial normalized so that the coe?cient of the leading term is unity. Our interest here is the action of U + on the kernel KA (x; y ; q, t). For this we require Theorem 5.2 (a), (b) and (5.8) as well as the result of the following lemma. Lemma 5.3 De?ne

n

(1 ? q )E0,m :=

i=m

Ai,m

(1 ? τi ) xi

n

where

Ai,m :=

j =m j =i

txi ? xj . xi ? xj

When acting on symmetric functions

n

Di = (1 ? q )E0,m

i=m

(5.9)

12

Proof.

When acting on symmetric functions, we see from (3.7) and (2.1) that

1 Dj = Tj . . . Tn?1 x? n (1 ? τn ).

(5.10)

1 In particular Dn = x? n (1 ? τn ) so (5.9) is true for m = n. Thus by induction (5.9) is equivalent to the statement that Dm?1 = (1 ? q )(E0,m?1 ? E0,m ) (5.11)

Noting that Ai,m?1 = 1 + (t ? 1) we see that (5.11) can be rewritten to read Dm?1 = Am?1,m?1

xi Ai,m xi ? xm?1

n (1 ? τi ) xi (1 ? τm?1 ) + (t ? 1) Ai,m := Rm?1 xm?1 x ? xm?1 xi i=m i

(5.12)

Since Rn = Dn , and from (5.10) Tj?1 Dj = Dj +1 , to establish (5.12) it su?ces to show

?1 Tm ?1 Rm?1 = Rm

(5.13) 2

This can be veri?ed by direct calculation using (5.12) and (2.8). We are now ready to calculate the action of U + on KA (x; y ; q, t). Proposition 5.4 We have U +(x) KA (x; y ; q, t) = [n]t ! 0 F0 (x; y ; q, t) where [n]t ! :=

n i=1 (1

(5.14)

n i=1 (i

? ti )/(1 ? t), and with κ denoting a partition and b(κ) := tb(κ) P (x; q, t)Pκ (y ; q, t). n?1 ; q, t) κ d′ κ (q, t)P (1, t, . . . , t

? 1)κi , (5.15)

0 F0 (x; y ; q, t) := κ

Proof.

Applying U + to the x-variables in (5.1) gives U +(x) KA (x; y ; q, t) =

η

dη (q, t)aη (q, t) Pη+ (x; q, t)Eη (y ; q ?1 , t?1 ) d′ ( q, t ) e ( q, t ) η η

(5.16)

Repeating this operation and use of Theorem 5.2 (a) shows U +(x) KA (x; y ; q, t) =

η+

αη+ (q, t)Pη+ (x; q, t)Pη+ (y ; q, t)

where we have used the fact [15] that Pη+ (y ; q ?1 , t?1 ) = Pη+ (y ; q, t). To specify αη+ sum Theorem 5.2 (b) over i, apply U +(x) to both sides of the resulting equation and commute its action to the right of i Di on the l.h.s. (Lemma 3.1 shows that this is valid), and use Lemma 5.3 to substitute for i Di to show (1 ? q )E0,1

(x) η+

αη+ (q, t)Pη+ (x; q, t)Pη+ (y ; q, t) = p1 (y )

η+

αη+ (q, t)Pη+ (x; q, t)Pη+ (y ; q, t)

(x)

Recent results of Lassalle [12, Theorems 3 and 5] give the action of the operator E0,1 on Pη+ (x; q, t)/Pη+ (1, . . . , tn?1 ; q, t) and an expression for the product p1 (y )Pη+ (y ; q, t)/d′ η (q, t) in terms of generalized binomial coe?cients. These formulas imply a recurrence for the quantity 13

n?1 ; q, t) with the unique solution α + d′ P + (1, . . . , tn?1 ; q, t) = tb(η ) α . αη+ d′ 0 η η+ Pη+ (1, . . . , t η+ η ? ( w ) + = [n]t ! gives α0 = [n]t !. 2 The fact that U 1 = w∈Sn t

+

As an application of Proposition 5.4 we can specify the proportionality constant aη in (5.8). For this we also require the formula [14, 16] Pκ (y ; q, t) = d′ κ (q, t) 1 d′ (q, t) η : η + =κ η Eη (y ; q, t), (5.17)

which is the analogue of (1.11). We ?rst substitute (5.16) on the l.h.s. of (5.14), then substitute . Next for Pη+ (y ; q, t) on the l.h.s. using Pη+ (y ; q, t) = Pη+ (y ; q ?1 , t?1 ) = RHS (5.17) ?1 ?1

q →q ,t→t

we note from the de?nition (2.17) that

?1 ?1 d′ η+ (q , t ) ?1 ?1 d′ η (q , t )

=

s∈η+ ,r ∈η

tl(r)?l(s)

d′ η+ (q, t) d′ η (q, t)

and equate coe?cients of Pη+ (x; q, t)Eη (y ; q ?1 , t?1 ) on both sides to conclude aη (q, t) = [n]t ! t

r ∈η

l(r )

eη (q, t) . Pη+ (1, . . . , tn?1 ; q, t)dη (q, t)

(5.18)

Substituting (5.18) in (5.8), we obtain the q -analogue of (1.12). The function 0 F0 (x; y ; q, t) appears in an unpublished manuscript of Macdonald [13], as well as the recent work of Lassalle [12]. Kaneko [8] introduced a similar function, with tb(κ) in (5.15) ′ replaced by (?1)|κ| q b(κ ) .

References

[1] T. H. Baker and P. J. Forrester. The Calogero–Sutherland model and polynomials with prescribed symmetry. solv-int/9609010, to appear in Nuc. Phys. B. [2] T. H. Baker and P. J. Forrester. Non–symmetric Jack polynomials and integral kernels. q-alg/9612003. [3] I. Cherednik. A uni?cation of the Knizhnik–Zamolodchikov and Dunkl operators via a?ne Hecke algebras. Inv. Math., 106:411–432, 1991. [4] C. F. Dunkl. Intertwining operators and polynomials associated with the symmetric group. Monat. Math. to appear. [5] C. F. Dunkl. Di?erential-di?erence operators associated to re?ection groups. Trans. Amer. Math. Soc., 311:167–183, 1989. [6] C. F. Dunkl. Integral kernels with re?ection group invariance. Canad. J. Math., 43:1213– 1227, 1991. [7] C. F. Dunkl. Hankel transforms associated to ?nite re?ection groups. In D. St. Richards, editor, Contemp. Math., volume 138, pages 123–138, 1992. ? Norm. Sup. 4e [8] J. Kaneko. q -Selberg integrals and Macdonald polynomials. Ann. Sci. Ec. s? erie, 29:1086–1110, 1996. [9] A. N. Kirillov and M. Noumi. A?ne Hecke algebras and raising operators for Macdonald polynomials. q-alg/9605004. 14

[10] F. Knop and S. Sahi. A recursion and combinatorial formula for Jack polynomials. qalg/9610016. [11] L. Lapointe and L. Vinet. alg/9607025. Rodrigues formulas for the Macdonald polynomials. q-

[12] M. Lassalle. Coe?cients binomiaux g? en? eralis? es et polyn? omes de Macdonald. preprint. [13] I. G. Macdonald. Hypergeometric functions. Unpublished manuscript. [14] I. G. Macdonald. A?ne Hecke algebras and orthogonal polynomials. S? eminaire Bourbaki, 47` eme ann? ee, Publ. I. R. M. A. Strasbourg, 797, 1994-95. [15] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, Oxford, 2nd edition, 1995. [16] K. Mimachi and M. Noumi. A reproducing kernel for nonsymmetric Macdonald polynomials. q-alg/9610014. [17] S. Sahi. A new scalar product for nonsymmetric Jack polynomials. Int. Math. Res. Not., 20:997–1004, 1996.

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