Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 30 April 2014

Notes and References

This is an excerpt of the paper Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory by Ivan Cherednik, in collaboration with Etsuro Date, Kenji Iohara, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Masatoshi Noumi, and Yoshihisa Saito.

Isomorphism theorems for the QAKZ equation

Let us now turn to the q-deformations. We introduce the quantum affine Knizhnik-Zamolodchikov (QAKZ) equation, and show that there is an isomorphism between solutions of the QAKZ equation and solutions of the generalized Macdonald eigenvalue problem.

Affine Hecke algebras and intertwiners

In this section we recall the definition of the affine Hecke algebra nt in the case of GLn.

Let t* be a parameter. Then nt is the algebra defined over by the following set of generators and relations: generators : T1,,Tn-1, Y1,,Yn, relations : (Ti-t) (Ti+t-1) =0,(1in-1) (4.1) TiTi+1Ti= Ti+1TiTi+1, (1in-2) (4.2) TiTj=TjTi, (|i-j|>1) (4.3) YiYj=YjYi, (1i,jn) (4.4) YiTj=TjYi, (ji,i-1), (4.5) Ti-1Yi Ti-1= Yi+1(1in-1). (4.6) The relations (4.1) are called the quadratic relations, (4.2)-(4.3) the Coxeter relations, (4.4) the commutativity, and (4.5),(4.6) the cross relations.

Set P=T1Ti-1 YiTi-1 Tn-1-1. It follows from the defining relations (4.1)-(4.6) that the right hand side is independent of i (1in) and therefore equals to P=T1Tn-1 Yn=Y1T1-1 Tn-1-1. (4.7)

Lemma 4.1. The algebra nt can be presented as nt= T1,,Tn-1,P /, (4.8) where the quotient is by the quadratic relations (4.1), the Coxeter relations (4.2)-(4.3) and the following:

(a) PTi-1=TiP (1<i<n),
(b) Pn is central.


Notice that in terms of Yi's we have Pn=Y1Yn. The relations (a) and (b) readily follow from (4.7) and the defining relations (4.1)-(4.6). For instance, PT1P-1=Y1 T1-1 (T2-1T1T2) T1Y1-1=Y1 T1-1 (T1T2-1T1-1) T1Y1-1=T2.

To establish (4.8), we start with T1,,Tn-1,P and introduce the elements Y1,,Yn by Y1=PTn-1T1, Y2=T1-1Y1 T1-1,. We must check the commutativity Y1Y2=Y2Y1, TjY1=Y1Tj (j>1), etc. using (a), (b). The first reads Y1T1-1 Y1T1-1= T1-1Y1 T1-1Y1. We plug in the above formula for Y1 and move P to the left. The commutativity with 'distant' T is obvious. The other relations formally follows from these ones. We leave the verifications to the reader as an exercise.

The QAKZ equation

In this section, we introduce the QAKZ equation.

Definition 4.1. For u, we define the intertwiners by Fi(u)= Ti+t-t-1eu-1 t+t-t-1eu-1 . (4.9)

They satisfy Fi(u)Fi (-u)=1, (4.10) Fi(u)Fi+1 (u+v)Fi(v) =Fi+1(v) Fi(u+v) Fi+1(u). (4.11) The second relation can be deduced from Lemma 3.7 as we did for the degenerate Hecke algebra.

The quantum affine Knizhnik-Zamolodchikov (QAKZ) equation is the following system of difference equations for a function Φ(v) that takes values in nt (or any nt-module). Φ(v1,,vi+h,,vn) = Fi-1 (vi-vi+1+h) F1(vi-v1+h) T1Ti-1Yi ×Ti-1 Tn-1-1 Fn-1 (vi-vn) Fi(vi-vi+1) ×Φ (v1,,vi,,vn) (i=1,,n). (4.12) Here h is a new parameter.

Theorem 4.2. The QAKZ system (4.12) is self-consistent. It is invariant in the following sense: if Φ(v) is a solution, then so is Fi(vi+1-vi)si Φ(v)=si (Fi(vi-vi+1)Φ(v)).

This follows from (4.10), (4.11). Later we will make it quite obvious.

Let us discuss the quasi-classical limit of the QAKZ system. Setting t=ekh/2= qk,q=eh, let h0. The generators Ti,Yi are supposed to have the form Ti = si+kh2+ (si2=1), Yi = 1+hyi+, where by we mean terms of order h2. The relations of the degenerate affine Hecke algebra for si,yi can be readily verified. Using the formula tTi-1Fi(u) =1+kheu-1 (si-1)+, we find that h-1 ( Φ(,vi+h,)- Φ(,vi,) ) = { yi+k ( j(>i) sij-1 evi-vj-1 -j(<i) sij-1 evj-vi-1 +i-n+12 ) } Φ(,vi,)+. Hence the AKZ equation (3.42) is a semi-classical limit (h0) of the QAKZ equation.

To make the QAKZ equations more transparent, let us discuss the action of the affine Weyl group. The affine Weyl group of type GLn is the semi-direct product 𝕊n=𝕊n n, where n=i=1n γi is a free abelian group of rank n. Define the action of 𝕊n on a vector v=(v1,,vn)n by sijv= (v1,,vj,,vi,,vn) =sjiv,i<j, γiv= (v1,,vi+h,,vn), γi(vj) =vj-hδij. We also introduce π=γ1s1 sn-1=s1 sn-1γn. Its action on n and the coordinates reads as πv=(vn+h,v1,,vn-1), πvn=v1-h, πv1=v2,.

Lemma 4.3. 𝕊n can be presented as 𝕊n= s1,,sn-1,π/, where the relations are si2=1,si sj=sjsi (|i-j|>1), sisi+1si =si+1sisi+1, and

(a) πsi-1π-1=si (1<i<n),
(b) πn is central.

It is convenient to represent the elements γi,π graphically.

Fig.7 shows a reduced decomposition of γi: γi=si-1 s1πsn-1 si.

For a function Ψ(v) with values in nt, let si(Ψ) = Fi(vi+1-vi) siΨ, (4.13) π(Ψ) = PπΨ. (4.14)

Theorem 4.4 ([Che1992-2]). The formulas (4.13), (4.14) can be extended to an action of 𝕊n.

vi+h γi vn vi+1 vi vi-1 v1 π vn+h v1 vn-1 vn vn-1 v1 Figure. 7. Graphs forγandπ

We denote this action by 𝕊nw:Ψw(Ψ). For instance, γi(Ψ) (v1,,vn)= Fi-1 (vi-1-vi)-1 F1(v1-vi)-1 PFn-1 (vi-vn-h) Fi(vi-vi+1-h) ×Ψ(v1,,vi-h,,vn). Hence the QAKZ equation simply means the invariance of Φ(v) with respect to the pairwise commuting elements γi: QAKZγi(Φ) =Φ(i=1,,n). (4.15)

Let us connect QAKZ with the q-KZ introduced by Smirnov and Frenkel-Reshetikhin [Smi1986, FRe1991]. We fix an N-dimensional complex vector space V and introduce TEnd(VV) by T=(t-t-1) i<jEii Ejj+ij EijEji +ti=1N EiiEii due to Baxter and Jimbo. The algebra nt acts on Vn by Ti(a1an) = a1T (aiai+1) an, (4.16) Pi(a1an) = Cana1 an-1, (4.17) where aiV and C=diag(λ1,,λn). One can check that this action is well-defined by a direct calculation.

For N=n, let (Vn)0=span { ew(1) ew(n)| w𝕊n } be the 0-weight subspace. Here e1,,en denote the standard basis of V. It is easy to see that this subspace is closed under the action of nt. We state the next proposition without proof.

Proposition 4.5. If N=n and λ=(λ1,,λn) is generic, then the 0-weight space (Vn)0 is isomorphic to Iλ=Ind[Y1,,Yn]nt(λ).

Writing down AQKZ in (Vn)0 we get the q-KZ (for GLn and in the fundamental representation). Combining this observation with the isomorphism with the Macdonald eigenvalue problem (our next aim) we can explain why the Macdonald polynomials appear in many calculations involving the vertex operators.

The monodromy cocycle

Let Φ be a solution of the QAKZ equation. Thanks to (4.15), w(Φ) is also a solution of the QAKZ equation for any w𝕊n. We define 𝒯wnt by w𝒯w=Φ-1 w(Φ)for w𝕊n and call it the monodromy cocycle. It follows From (4.13) and (4.14) that Fi(vi-vi+1) Φ=siΦ𝒯i (4.18) and PΦ=π-1 Φ𝒯π. (4.19) Here 𝒯i stands for 𝒯si.

Lemma 4.6. w2-1 (𝒯w1) 𝒯w2=𝒯w1w2 forw1,w2 𝕊n.

Indeed, Φw1w2 𝒯w1w2= w1w2˜ (Φ)=w1 (w2Φ)= w1 (Φw2𝒯w2) =Φw1𝒯w1 w1w2𝒯w2.

The QAKZ equation implies that 𝒯γi=1. Hence 𝒯w depends only on the image w of w in 𝕊n.

Let (n,nt) be the set of nt-valued function on n. Next we define two anti-actions of 𝕊n: σw(Ψ) = w-1Ψ, (4.20) σw(Ψ) = w-1Ψ 𝒯w, (4.21) where w𝕊n and Ψ(n,nt). Lemma 4.6 means exactly that σ is an anti-action σw1w2=σw2σw1). For instance, σγi(vi)=vi+h=σγi(vi).

We note that in the difference theory the monodromy can be always made trivial. Indeed, the 1-cocycle {𝒯w,wW} is always a co-boundary because of the Hilbert 90 theorem. Hence conjugating solutions of AQKZ we can always get rid of the monodromy. So the above actions σ,σ are not too much different in contrast to the differential theory.

This argument can be applied to the AQKZ itself, although the group n is infinite. We can formally solve the QAKZ equation as follows. Let Ψ(n,nt). Then the infinite sum bBb (Ψ), (4.22) where B=i=1nγi𝕊n, satisfies the AQKZ, provided the convergence. For example, if Ψ is rapidly decreasing, then one can check that bBb(Ψ) is convergent.

We see, that constructing End(V)-valued solution Φ to QAKZ for finite-dimensional nt-modules V poses no problem. What is more difficult is to ensure a proper asymptotical behavior.

Isomorphism of QAKZ and the Macdonald eigenvalue problem

In this subsection, we introduce the Macdonald eigenvalue problem and prove its equivalence to the QAKZ equation. This is a q-analogue of the relation between AKZ and QMBP discussed in §3.4.

Let Φ be a solution of the QAKZ equation with values in End(V) for a nt-module V. We assume that it is invertible for sufficiently general v. Setting σi=σsi, we get from (4.18) and (4.9): Fi(vi-vi+1) Φ=σi(Φ), TiΦ= ( tσi+ t-t-1 evi-vi+1-1 (σi-1) ) Φ. Let us introduce the operator Tˆi (1in) by Tˆi=t σi+ t-t-1 evi-vi+1-1 (σi-1). (4.23) Then TˆiΦ=TiΦ and σπΦ=PΦ (see (4.19)). The operators Tˆi and σπ commute with the left multiplication by Tj, P and any elements from nt. Using all these: YiΦ = Ti-1-1 T1-1P Tn-1Ti+1 TiΦ = Ti-1-1 T1-1P Tn-1Ti+1 TˆiΦ = TˆiTi-1-1 T1-1P Tn-1Ti+1 Φ = Tˆi Tˆn-1 σπ (Tˆ1)-1 (Tˆi-1)-1 Φ. We come to the following definition: Δi= Tˆi Tˆn-1 σπ (Tˆ1)-1 (Tˆi-1)-1 ,1in. (4.24)

Since YiΦ=ΔtΦ and Yi commute with each other, [Δi,Δj] =0. By the construction, the operators Δi act in End(V)-valued functions. However if we understand them formally the commutativity can be deduced from the relations σivi= vi+1σi, (4.25) σiσγi= σγi+1σi. (4.26) σγi= σγi (4.27) The latter means that 𝒯γi=1.

Let Q be a polynomial in n variables. Then Q(Y1,,Yn)Φ =Q(Δ1,,Δn) Φ and we can represent Q(Δ1,,Δn) =w𝕊n Dw(Q)σw, (4.28) where Dw(Q) are pure difference operators, which do not contain σw (w𝕊n).

For symmetric Q, we introduce a difference operator of Macdonald type MQ by MQ=w𝕊n Dw(Q).

Let φ be a -valued function on n. The system MQφ=Q (λ1,,λn)φ (4.29) will be called the Macdonald eigenvalue problem. The operators MQ can be calculated for σ instead of σ. As in the differential case, the result will be the same.

Fix λ=(λ1,,λn)n. We take a left nt-module Vλ with the following properties:

(1) for any symmetric polynomial Q in n variables and all aVλ, Q(Y1,,Yn)a= Q(λ1,,λn)a,
(2) there exists a -linear map tr:Vλ such that tr((Ti-t)a)=0 for all i and aVλ.

As always, we fix a (local) invertible solution Φ(v) of the QAKZ equation with values in End(Vλ). Note that all Vλ-valued solutions of the QAKZ equation can be written in the form φ(v)=Φ(v)a(v) for B-periodic Vλ-valued function a(v): a(,vi+h,) =a(v)fori=1, ,n.

Theorem 4.7. Let Vλ be an nt-module with the above properties, 𝒮olQAKZ(Vλ) be the space of solutions of the QAKZ equation with values in Vλ, and 𝒮olMac(λ) the space of solutions of the Macdonald eigenvalue problem (4.29). Then 𝒮olQAKZ(Vλ) tr𝒮olMac(λ).


Let φ(v)=Φ(v)a𝒮olQAKZ(Vλ). Then (σi-1)Φ= (t+t-t-1evi-vi+1-1)-1 (Ti-t)Φ. For a reduced decomposition w=si1sil of w𝕊n, σw-1 = σil σi1-1 = σsil σsi2 (σsi1-1) +σil σi2-1 = k=1l σil σik+1 (σik-1). Since σi commutes with the left action of {T}, we have (σw-1)Φ = k=1l σil σik+1 (σik-1) Φ = k=1l σil σik+1 (a scalar function)(Tik-t) Φ = k=1l(a scalar function) (Tik-t) σil σik+1. Φ Using the commutativity of Dw(Q) with Ti-t we represent Dw(Q)(σw-1)Φ as a sum (Ti-t)Ψi for some nt-valued functions Ψi. Finally Q(λ1,,λn) Φ = Q(Δ1,,Δn) Φ = w𝕊n Dw(Q) σwΦ = w𝕊n Dw(Q)Φ +w𝕊n Dw(Q) (σw-1)Φ = MQΦ+ (Ti-t)Ψi. Applying this relation to aVλ and taking tr, we conclude: Q(λ1,,λn) tr(φ)=MQtr(φ).

Let us now consider Vλ=Jλ. The definition is quite similar to the differential case. We start with Jλ=IndHntnt (+)/Lλ. Here Hnt=T1,,Tn-1nt, +:Hnt is the one-dimensional representation sending Ti to t, and Lλ is the ideal generated by p(Y1,,Yn)-p(λ) (p[x1,,xn]𝕊n). As in §3.1, Jλ stands for the dual module defined via the anti-involution of nt: Yi=Yi, Ti=Ti.

The main result of this subsection is the following theorem from [Che1992, Che1994-2].

Theorem 4.8. If Vλ=Jλ, then the map from 𝒮olQAKZ(Vλ) to 𝒮olMac(λ) is injective.

The theorem results from the following two lemmas.

Lemma 4.9. Let K be a nt-submodule of Jλ. Then tr(K)=0 implies K=0.

The proof repeats that in the differential case (see ((3.38))).

Lemma 4.10. Let φ be a Vλ-valued solution of the QAKZ equation. Assume tr(φ)=0. Then tr(ntφ)=0.


First tr(Tiφ)=t tr(φ)=0 for all i. Then π=s1s2sn-1γn, σπ-1 = σγn σn-1 σ1-1 = σγn ( k=1n-1 σn-1 σk+1 (σk-1) ) +σγn-1, and tr(σγnφ)=tr(φ)=0. Therefore, representing φ=Φa (aVλ), we have tr(Pφ)-tr(φ) = tr((σπ-1)Φa) = tr ( σγn ( k=1n-1 σn-1 σk+1 (σk-1)Φ ) a ) = k=1n-1tr ( σγnσn-1 σk+1fi (v)(Tk-t)Φa ) = k=1n-1tr ( (Tk-t) σγnσn-1 σk+1fi (v)Φa ) = 0, where fi(v) are -valued function. Hence tr(Pφ)=0 and tr(Ynφ) = tr(Tn-1-1T1-1Pφ) = t1-ntr(Pφ) = 0.

Now we shall prove that tr(Yiφ)=0 for all i by induction. Assume that tr(Yiφ)=0 for k+1in. Since Yk=Tk-1-1T1-1PTn-1Tk it is enough to see that tr(PTn-1Tkφ)=0. Since φ is a solution of the QAKZ equation we have tr ( Fk-1-1 F1-1PFn-1 Fkφ ) = tr(γk-1φ) = γk-1tr(φ) = 0. On the other hand, Fi(v)=ci (v)(Ti+fi(v)) where ci(v) and fi(v) are some scalar functions. Therefore 0=tr (PFn-1Fkφ) = tr ( cn-1ckP (Tn-1+fn-1(v)) (Tk+fk(v)) φ ) = I=(i1,,il) tr(cI(v)PTilTi1φ) where I=(i1,,il) is a sequence of integers such that ki1<i2<<iln-1, and cI(v) is some scalar function. If II0=(k,k+1,,n-1) then there are the following possibilities:

(1) iln-1,
(2) il=n-1 and there exists an m (1ml) such that ij-ij-1=1 for any j=m+1,m+2,,l and im-im-1>1,
(3) otherwise.

case (1): As il<n-1, we have tr(PTilTi1φ) = tr(Til+1Til+1Pφ) = tltr(Pφ) = 0.

case (2): Since [Ti,Tj]=0 for |i-j|>1, tr ( P(TilTim) (Tim-1Ti1) φ ) = tr ( P(Tim-1Ti1) (TilTim)φ ) = tr (Tim-1+1Ti1+1 PTilTimφ ) = tm-1tr (PTilTimφ). By the induction hypothesis, tr(PTilTimφ)=0. Hence tr(PTilTi1φ)=0.

case (3): In this case I=(i1,,il) must be of the form il=n-1, il-1=n-2, , i1=n-l>k. By induction, tr(PTilTi1φ)=0. So tr(Yiφ)=0 for all i.

Because of the relations between T and Y, it remains to check that tr(Yi1Yilφ)=0 for any l. One can show this by induction on l.

Macdonald operators

We set Tˆi = tσi+ t-t-1evi-vi+1-1 (σi-1), (1in-1), (4.30) Gij = t+ t-t-1evi-vj-1 (1-σij), (1i,jn), (4.31) Δi = Tˆi Tˆn-1σπ Tˆ1-1 Tˆi-1-1. (4.32) Here σw are from (4.20), σij=σsij.

Switching from {T} to {G}: Tˆiσi = Gii+1, Gij-1 = t-1- t-t-1evi-vj-1 (1-σij), Δi = Gii+1 Ginσγi G1i-1 Gi-1i-1. Let em be the m-th elementary symmetric polynomial in n variables. We represent em(Δ1,,Δn)= w𝕊nDw(m) σw, (4.33) for difference operators Dw(m), and define Mm=Mem= w𝕊n Dw(m). All these operators are W-invariant, which results from the following lemmas.

Lemma 4.11. Consider the algebra ˆ generated by Tˆi (1in-1), Δj (1jn). Then TiTˆi, YjΔj extends to an algebra isomorphism ntˆ. Moreover, if Q is a symmetric polynomial in n variables, then Q(Δ1,,Δn) is a central element in ˆ.

Actually this observation is the key point (it can be checked directly or with some representation theory). We note that the formulas for T generalize the so-called Demazure operations and the Bernstein-Gelfand-Gelfand operations. They were also studied by Lusztig and in a paper by Kostant-Kumar.

From now on we identify ˆ with nt.

Lemma 4.12. Let f(v1,,vn) be a function on n. Then f is symmetric if and only if (Tˆi-t)f=0 for all i.

Lemma 4.13. Let Q be a symmetric polynomial in n variables. Then Q(Δ1,,Δn) acts on the space of the symmetric polynomials in evi (1in).


This follows immediately from Lemma 4.11 and 4.12.

Let us calculate M1. Since M1 is symmetric, it is enough to find the coefficient of σγ1. Using the G-representation it is easy to see that σγ1 does not appear in Δ2.,Δn. The σγ1-factor of Δ1 is equal to i=2ntev1-vi-t-1ev1-vi-1σγ1.

After the symmetrization we get the formula: M1=i=1n ji tevi-t-1evj evi-evj σγi.

Similarly, Mm=I=(i1,,im) iI,jI tevi-t-1evj evi-evj σγi1 σγim where I=(i1,,im) is a sequence of integers such that 1i1<<im<n.

To recapitulate, let us consider the classical limit of the Macdonald operators. Setting q=eh and t=qk/2, h0, we have Δi = 1+h𝒟i+O(h2), M1-n = hvi+ O(h2), M2-(n-1)M1+ n(n-1)2 = h2 ( L2+(n+13) k3 ) +O(h3).


Remark 4.1. Take a solution Φ=Φ(v) of the QAKZ equation in an t-module V, assuming that Φ has the trivial monodromy. Then, for any polynomial p[x1,,xn], we have p(Y1,,Yn) Φ=p(Δ1,,Δn) Φ, (4.34) where Δi are the difference Dunkl operators defined before. Note that Δi can be replaced by Δi because the monodromy of Φ is trivial. We also need a linear functional pr:Vλ for a vector λ=(λ1,,λn)n such that pr(Yib)=λi pr(b)(i=1,,n) (4.35) for any bV. Given any element aV, let us define a scalar-valued function φ=φ(v) setting φ(v)=pr (Φ(v)a). (4.36) Then the formula (4.34) implies p(λ1,,λn) φ=p(Δ1,,Δn) φ. (4.37) Thus, the scalar-valued function φ=φ(v) solves the Dunkl eigenvalue problem.

Remark 4.2. Arbitrary root systems. Let Σ={α}n be any reduced root system of rank n (of type A, B, C, D, E, F or G), and t= T1,,Tn, X1,,Xn (4.38) the corresponding affine Hecke algebra. The baxterization (a parametric deformation satisfying the Yang-Baxter relations) of Ti will be given by Fi=Ti+ t-t-1 eui-1 withui= (u,αi) (4.39) for each i=1,,n. We also have to use the element T0=Xθ Tθ-1 (4.40) corresponding to the simple affine root α0=δ-θ for θ being the highest root. Its baxterization is quite similar: F0=T0+ t-t-1 eh-uθ-1 , (4.41) where uθ=(u,θ). The functions F0,F1,,Fn satisfy the Yang-Baxter equations associated with the extended Dynkin diagram. For example, in the case of 12, we have F1(v) F2(u+v) F1(2u+v) F2(u)= F2(u) F1(2u+v) F2(u+v) F1(v). (4.42) The arguments of Fi can also be determined graphically by means of the equivalent pictures of the reflection of two particles (see [Che1992-2]).

Using T0, the affine Hecke algebra t has an alternative representation t= T0,T1,,Tn;Π, (4.43) where Π is a certain finite abelian group. The group Π is isomorphic to P/Q. It is the set of all elements of the extended affine Weyl group W=WB,B= i=1nbi, (4.44) preserving the set {α0,α1,,αn} of the simple affine roots. It gives the embedding of Π into the automorphism group of the extended Dynkin diagram. The action of W on nδ is by the affine reflections and the corresponding shifts in the δ-direction for B: b(z+ζδ)=z+ (ζ-(b,z))δ.

Lemma 4.14. W=s0,s1,,sn;Π with s0=(θ)·sθ.

The group Π can be embedded into the affine Hecke algebra. The images Pπ of the elements πΠ permute {Ti} in the same way as π do in W with {si}. The baxterization of the elements in Π is trivial: Fπ=Pπ for each πΠ.

Keeping the notations of the previous sections, we have the following theorem.

Theorem 4.15. Given any t-valued function Ψ=Ψ(u), the formulas si(Ψ)= si(FiΨ) (4.45) for all i=0,1,,n, and π(Ψ)= PππΨ (4.46) for all πΠ induce a representation of W.

The QAKZ equation for Σ is the invariance condition b(Φ)=Φ for all bB. It can be shown that this equation is equivalent to the difference QMBP associated with the root system Σ defined via similar Dunkl operators. A conceptual proof of this isomorphism theorem is given by means of the intertwiners of double affine Hecke algebras (see [Che1992, Che1994-2]).

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