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: 28 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 AKZ equation

We introduce the affine Hecke algebra Σt and connect them with the degenerate affine Hecke algebra Σ using the monodromy of the AKZ equation. We also establish an isomorphism between the solution space of the AKZ equation and that of a quantum many body problem.

Representations of Σ

In this section we define induced representations of Σ.

For λ=(λ1,,λn)n, the character of [x1,,xn] (i.e. a ring homomorphism [x1,,xn]) is an assignment xiλi. We denote it by λ.

Definition 3.1. We define an Σ-module Iλ as the representation induced from λ: Iλ= Ind[x1,,xn]Σ (λ)=Σ [x1,,xn] λ. (3.1) Here λ is endowed with the [x1,,xn]-module structure by the character λ.

We have the Poincaré-Birkhoff-Witt type theorem for Σ. Namely any hΣ is expressed uniquely in either of the following ways: h=wWpw (x)w=wW wqw(x) (3.2) with pw,qw[x1,,xn]. The existence results from the relations (2.32)-(2.34) in Σ. Hence Iλ=[W]= wWw. (3.3) Thus Iλ is [W] as a W-module, where the action of xi is determined by xi(e)=λie for the identity eW. The action of xi on other elements of [W] have to be determined using the defining relation (similar to the calculations in the Fock representation).

We also need another construction. Let J be induced from the trivial character +:W, w1. Then J=Ind[W]Σ (+), (3.4) is isomorphic to [x1,,xn] as a vector space and moreover as a [x1,,xn]-module. To get finite-dimensional representations from J, we use the coincidence of the center of Σ with the algebra of W-invariant polynomials in xi. This theorem is due to Bernstein. The procedure is as follows. Let us fix an element λ=(λ1,,λn)n and introduce the ideal Lλ in [x1,,xn] generated by p(x)-p(λ) for all W-invariant polynomials p. Set Jλ=J/Lλ. Then Jλ has a structure of Σ-module by virtue of the Bernstein theorem.

We will also use the anti-involution ˚ on Σ: xi=xi, si=si, (ab)=b a,k=k. (3.5) Since the relations of Σ are self-dual it is well-defined. For an Σ-module V, we consider its dual Hom(V,). The dual has an anti-action (a right action) of Σ. Composing it with the anti-automorphism ˚, we get a natural (left) action of Σ. We denote the resulting module by V.

We write λb=kiλi for b=kibi.

Theorem 3.1.

(a) Iλ is irreducible if and only if λα±k for any αΣ+.
(b) There exists a permutation λ of λ (i.e. λ=w(λ) for wW) such that λα-k
 for any αΣ+. Then JλIλ. (3.6)
(c) For the longest element w0 in W, Iλ= Iw0(λ) (3.7)

A key lemma in proving Theorem 3.1 is

Lemma 3.2. I(0,,0) is irreducible.

The proof from [Che1995-2] is based on the intertwining operators of degenerate affine Hecke algebras (to be defined below). See also [KLu0862716, Kat1981, Rog1985] and the references therein (the non-degenerate case).

Definition 3.2. For 1in we set fi=fsi= si-kxai. (3.8) For wW with a reduced decomposition w=sinsi1, fw=finfi1. We call the elements fw intertwiners.

The elements fw belong to the localization of the degenerate affine Hecke algebra Σ by the W-invariant polynomials. They give a certain 'baxterization' of w, and are closely related to the Yang's R-matrix. Let us show that fw does not depend on the choice of the reduced decomposition of w.

We have fsixb= xsi(b) fsi, (3.9) fwxb= xw(b) fw. (3.10) Indeed, (si-kxαi) xb=xsi(b) (si-kxαi), (3.11) which can be rewritten as follows: sixb- xsi(b) si=-k xsi(b)-xbxαi. (3.12) Using the definition of xb (2.36), the right hand side of (3.12) is k(b,αi). So we come to (2.37). The relations (3.10) fix fw uniquely up to the multiplication on the right by functions in x. The leading terms of fw being w, they coincide for any reduced decompositions.

To demonstrate the role of intertwiners, let us check the irreducibility of Iλ for generic λ. First note that the vectors {fw(e)Iλ} are common eigenvectors of xb, because xbfw(e)= fwxw-1(b) (e)=λw-1(b) fw(e), For a generic λ the eigenvalues are simple, hence these vectors are linearly independent. Now, any nonzero Σ-submodule A of Iλ contains at least one eigenvector of xb. By the simplicity of eigenvalues, such an eigenvector must be in the form fw(e) for some wW. On the other hand fw are invertible elements. Indeed, fi-1= (1-k2xαi2)-1 fi. Therefore eA. Since Iλ is generated by e, we conclude that A=Iλ.

Actually this very reasoning leads to the proof of the Theorem (a),(b). However if λ is arbitrary one must operate with the intertwiners much more carefully. It is necessary to multiply them by the denominators and remember that the invertibility does not hold for special λ.

Remark 3.1. The -quotients A of Jλ will be interpreted below as certain quotients of the D-module representing the quantum many-body eigenvalue problem. A solution of the AKZ in Jλ induces solutions in any of its -quotients (if I is reducible). It gives a one-to-one correspondence between the -submodules (quotients, constituents) of J and those of the D-modules representing the quantum many-body eigenvalue problem. The description of the latter is an analytical problem. The classification of the former is a difficult question in the representation theory of Hecke algebras. For instance, the multiplicities of the irreducible constituents are described in terms of the Kazhdan-Lusztig polynomials. It is very interesting to combine the two approaches together.

The monodromy of the AKZ equation

In this section we discuss the monodromy of the AKZ equation, which is a key ingredient in establishing the isomorphism between the AKZ equation in the representation Jλ and the quantum many-body problem (QMBP) with the eigenvalue λ.

Let U be the open subset of n given by U= { un| αΣ+ (euα-1) 0 } . (3.13) The lattice generated by b1,,bn will be denoted by B. It is isomorphic to n and acts on n by translations. Namely, b(u)=u+2π-1b, where bB. The semi-direct product W=WB is the so-called extended affine Weyl group, acting on n and leaving U invariant. Picking u0U, we set π1=π1 (U/W,u0). The group structure of π1 is described as follows. Given an element wW, let γw be a path from u0 to w-1(u0) in U. For elements w1,w2W, we define the composition γw2γw1 of γw1 and γw2 as the path composed of γw1 and the path γw2 mapped by w1-1 (see Fig.5). The class of γ will be denoted by γ. The map γww is a homomorphism onto W.

It is convenient to choose u0 and the generators of π1 as follows. Set =Re, =Im, C=(-1)n\ { u(-1)n |0<uα <2πfor everyα Σ+ } . Then n\C is a simply connected open subset of U. Let us take u0n such that uα00 for αΣ+. For any element wW, we denote a path from u0 to w-1(u0) in n\C by γw. This condition simply means that whenever uαi intersects the imaginary axis it must go through the 'window' 0<uα<2π. u0 γw1 w1-1(u0) γw2 w1-1 w2-1(u0) w1-1w2-1(u0) Figure. 5. Composition of paths

For any element wW we define an element γw of π1 to be the image of γw. Since n\C is simply connected, γw depends only on w. We set τi=γsi and choose χi to be a path from u0 to the point u with the same coordinates uj=uj0 for ji and ui=ui0+2π-1. The structure of π1 is described in the following theorem from [Lek1981].

Theorem 3.3. π1= τ1,, τn, χ1,, χn , (3.14) τi satisfy the Coxeter relations, (3.15) [χi,χj]= [τi,χj]=0 (ij). (3.16) τi-1 χi τi-1 =χsi(bi). (3.17) Here for b=i=1nkibi we put χb= i=1n χiki. (3.18)

Fig.6 proves the relation (3.17). It shows the ui-coordinate only, which is sufficient for this relation.

Let us introduce the affine Hecke algebra Σt associated with a root system Σ as a quotient of the group algebra of π1 by the quadratic relations.

Definition 3.3. The affine Hecke algebra associated with a root system Σ is an associative -algebra generated by 1, T1,,Tn, X1,,Xn with the following relations: Tisatisfy the Coxeter relations. (3.19) [Xi,Xj]= [Ti,Xj]=0 ij, (3.20) Ti-1Xi Ti-1= Xsi(bi), (3.21) (Ti-t) (Ti+t-1) =0. (3.22) 0 2π-1 χsi(bi) χi si(u0) u0 τi τi si(u0)+2π-1bi u0+2π-1bi Figure. 6. Proof of the relation (3.17) The monomials Xb are defined as in (3.18), t*. Here and above we mean the homogeneous Coxeter relations: TiTjTi=TjTiTj, mij factors on each side, where mij=2,3,4 whenever the corresponding vertices in the Dynkin diagram are connected by 0,1,2 laces.

Let Φ be an invertible solution of the AKZ equation associated with Σ, defined in a neighborhood of u0. Then, for wW, w-1(Φ) is defined near w-1(u0) (see (2.30)). Let γ be a path in U from u0 to w-1(u0). Denote by (w-1(Φ))γ the analytic continuation of w-1(Φ) back to u0 along the path γ, where γ denotes the class of γ in the fundamental group π1. We will also use the projection homomorphism WW sending w=wb to w for bB, wW. Using this homomorphism we can extend the action of W from (2.30) to W, multiplying Φ on the left by w.

Let us define the monodromy Tγ to be the ratio Tγ= (w-1(Φ))γ-1 Φ=(Φ(w(u)))γ-1 ·w·Φ. (3.23) Here dot means the product in Σ. Since Φ and w-1(Φ) both satisfy the same AKZ equation, Tγ does not depend on u. So it is an invariant of the homotopy class of γ and is always invertible. If we choose u0 and the paths γw in n\C as above, then Tw for wW are well-defined. The monodromy is a homomorphism from π1 (but not from W), which readily results from the definition.

As a preparation for an explicit computation of {Tw} in the next section, we shall introduce a special class of solutions Φ.

Proposition 3.4. For generic λ, there exists a unique solution Φas(u) of the AKZ equation such that Φas(u) = Φˆ(u) ei=1nuixi for (3.24) Φˆ(u) = 1+m=(m1,,mn),mi0,m0 Φme-i=1nmiui, (3.25) where u, and Φm are independent of u.

We call the solution in the proposition the asymptotically free solution. To be more exact, we need either to complete Σ, or restrict ourselves with finite-dimensional representations of this algebra. Then establishing the the (local) convergence is easy. In these notes we will follow the second way. We give general formulas, which are quite rigorous in finite-dimensional representations (say, in the induced representations).

Let us examine the condition necessary for the existence of the asymptotically free solutions in the case of A1. A general consideration follows the same lines. In this case, Φas(u)= (1+m>0Φme-mu) eux=Φˆ (u)eux. (3.26) The equation (2.39) leads to Φˆ(u)u =kseu-1Φˆ (u)+[x,Φˆ(u)]. (3.27) Comparing the coefficients of e-mu: -mΦm= [x,Φm]+ (terms with Φj,j>m). (3.28) Given a representation of Σ, we find Φm assuming that m+ad(x) is invertible for any m>0 in this representation. Therefore, setting Spec(x)={μj}, the conditions m+μi-μj0, m=1,2,, ensure the existence of the asymptotically free solutions. The convergence estimates are straightforward. These conditions are fulfilled in generic induced representations.

Lusztig's isomorphisms via the monodromy

In this section we establish an isomorphism between Σt and Σ using the monodromy of the AKZ equation.

Let us fix an invertible solution Φ(u) of the AKZ system in a neighborhood of u0U*=n\CU. The functions Φ(w(u)) will be extended to u0 through U*. Since Σ is infinite-dimensional, we have to consider all formulas in finite dimensional representations. Once we get the final expressions it is not difficult to find a proper completion of the degenerate Hecke algebra for them.

Theorem 3.5 ([Che1991-4]). There exists a homomorphism from Σt to Σ given by TjTj, XjXj, where Tj=Φ (sj(u))-1 sjΦ(u), Xj=Φ (u-2π-1bj)-1 Φ(u). If t=exp(π-1k) is sufficiently general (say, not a root of unity), then it is an isomorphism at the level of finite dimensional representations or after a proper completion.

Under the notation (3.23), Tj=Tτj and Xj=Tχj. Hence the relations (3.19)-(3.21) result from Theorem 3.3, and only the quadratic relations (3.22) need to be proved. We skip a simple direct proof since these relations follow from the exact formulas below.

Let us find the formulas for Tj and Xj for the asymptotically free solution Φas(u). Given bB, we set Xb=j=1n Xjkjforb= j=1nkjbj, and define Xb analogously.

Theorem 3.6 ([Che1991-3]). Let us choose the asymptotically free solution Φas(u) as Φ(u). Then

(a) Xj=exp(2π-1xj),
(b) si-kxαi=g(xαi)(Ti+t-t-1Xa-1-1),
where the function g(v) is defined by g(v)= Γ(1+v)2 Γ(1+k+v) Γ(1-k+v) , and (b) is in fact a formula for Ti in terms of {s,x}.

We will give a sketch of the proof of Theorem 3.6. The statement (a) is immediate, since Φas(u-2π-1bj) =Φˆ(u) euixi-2π-1xj =Φas(u) e-2π-1xj. To prove the statement (b), we reduce the problem to the A1 case. Let us fix the index i (1in). Set E(u)=ei=1nuixi, so that Φas(u)=Φˆ(u)E(u). Let us define Φ(i)(u) as follows: Φ(i)(u)= Φ(i) (ui)E(u), where Φ(i)(ui)=limuj+(ji)Φˆ(u). The AKZ system for Φ(i)(u) reads: Φ(i)ui = (ksieui-1+xi) Φ(i), (3.29) Φ(i)uj = xjΦ(i) (ji). (3.30) Reduction procedure. Since the monodromy Ti does not depend on u, the point u0 and the path connecting u0 and si(u0) may be replaced by any deformations in U or their limits. Provided the existence, the resulting monodromy coincides with Ti. For instance, Ti equals Ti(i)= (Φ(i)(si(u)))-1 siΦ(i)(u).

Indeed, the latter is the limiting monodromy for a path with uj (ji) approaching the infinity. We note that uj(si(u0))+ if uj(u0) does.

In the reduced equations (3.29) and (3.30), we may diminish the values, considering the subalgebra of Σ generated by xj (1jn), and si. In this algebra, the following elements are central: xj(ji), xi-12xαi. Hence, if we define E(i)(u) by E(i)(u)= ej=1nujxj-uixαi/2, it enjoys the following properties:

(i) E(i)(u) commutes with si,
(ii) E(i)(si(u))=E(i)(u).
The second property can be verified directly: j(si(u))j xj-12(si(u))i xαi=j (uj-(ai,αj)ui) xj+12uixai =jujxj-12ui xai. We have used that (si(u))j=(αj,si(u)) and j(ai,αj)xj=xai. Setting Φ(i)(u)= Φ(i)(u)E(i) (u)-1, the system of equations (3.29),(3.30) becomes precisely the AKZ equation for Φ(i)(u) in the A1 case: Φ(i)(u)ui = ( ksieui-1 +12xai ) Φ(i)(u), (3.31) Φ(i)(u)uj = 0(ji). (3.32) Because of the above properties of E(i)(u), the monodromy of Φ(i)(u) coincides with Ti. However Φ(i)(u) can be expressed in terms of the hypergeometric function, which conclude the proof up to a straightforward calculation.

To explain the structure of the formula for T, let us involve the intertwiners of Σt. They are defined similar to those in the degenerate case: fi=si-kxai forΣ, Fi=Ti+ t-t-1Xai-1-1 forΣt.

Lemma 3.7. FiXb=Xsi(b)Fi.

It readily results from the definition of Σt (cf. 3.10).

The image Fi of Fi in Σ with respect to the homomorphism constructed in Theorem 3.5 can be represented as Fi=gi(x)fi for a function gi of x. Indeed, fiXb=Xsi(b)fi, which gives the proportionality. Recall that Xb=exp(2π-1xb). Here gi(x) must be of the form g(xai) for a function g in one variable, and can be calculated using the hypergeometric equation (3.31). We omit the details (see [Che1991-3]).

We note, that the quadratic relations for Ti can be made quite obvious using the same reduction (the exact formulas above are not necessary). Set i=1 to simplify the indices. We switch from (3.31) to (2.18) with two variables z1,z2 and a parameter z0: Φzj= [ k(s1zj-zk) +Ωjzj-z0 ] Φ(j=1,2,k=3-j). (3.33) When z0=0 the substitutions are as follows 2x1=Ω1-Ω2 +ks1,u1=log (z1/z2),Φ =Φ(1)(u1) (z1z2)-1/2(Ω1+Ω2+ks1). The monodromy corresponding to the transposition of z1 and z2 for z0=0 coincides with T1. It does not depend on z0 up to a conjugation (the same reduction argument applied to the KZ-equation with three variables). Sending z0 to infinity we eliminate the Ω-terms. The monodromy of the resulting equation can be calculated immediately. Since it is conjugated to T1 we get the desired quadratic relations.

Heckman in [Hec1987] used a similar reduction approach when calculating the monodromy of the quantum many-body problem (also called the Heckman-Opdam system). Our next aim is establishing an isomorphism of AKZ and the latter. Combining Heckman's formulas and mine for the AKZ, which coincide since the representation of t is the same, we readily conclude that these equations are isomorphic for generic λ. This will be made much more constructive below. We will also consider any λ.

Remark 3.2. Let us apply Theorem 3.6 to the standard rational KZ equation in the GLn case. We calculated the monodromy of Φvi= ( kj>i sijevi-vj-1 -kj<i sijevj-vi-1 +yi ) Φ(1in). Taking special yi=kj=i+1nsij and substituting zi=evi, we come to Φzi=k ji sijzi-zj Φ(1in). It corresponds to the simplest Ωij=0 in (2.18). By the way, these {y} induce a homomorphism from n to 𝕊n+1 due to Drinfeld. Diagonalizing the commuting elements j>isij we recover the monodromy computed by Tsuchiya-Kanie [TKa1988]. It also gives an explicit example of the general results on the monodromy of the rational KZ over Lie algebras due to Drinfeld and Kohno (see [Koh1987]).

Remark 3.3. In Theorems 3.5 and 3.6, we established the isomorphism Σt Σ, Xjt2xj, where t=eπ-1k and represented it as a relation between the intertwiners of the degenerate and non-degenerate affine Hecke algebras: Fj=Tj+ t-t-1Xaj-1-1 g(xaj) (sj-kxaj). This construction can be naturally generalized. In fact we need only a very mild restriction on g(x) to get such a homomorphism. Normalizing the intertwiners to make them 'unitary' (f2=1=F2), we come to the simplest possible map: Xjt2xj, Fjt+t-t-1Xaj-1-1 sj-kxaj1-kxaj.

Actually here we have four formulas in one since we can put the denominators on the right and on the left. One of them was found by Lusztig in [Lus1989].

The isomorphism of AKZ and QMBP

Here we present the isomorphism between the AKZ equation and the quantum many-body problem (QMBP). The latter will appear as a 'trace' of the first.

We will need a variant of the general notion of monodromy by A. Grothendieck. Let us fix the notations: wΦ(u)= Φ(w-1(u)) ,w=wbW =WB,un.

Given a finite union C of affine real closed half-hyperplanes, we set U=n\C assuming that

(i) U does not contain 'bad hyperplanes' αΣ+(euα-1)=0,
(ii) U is simply connected,
(iii) (n\wWw(C))/W is connected.
We shall refer to such C as a system of cutoffs. In §3.2, a special system of cutoffs (U*) has been already used in order to compute the monodromy.

Let us fix a system of cutoffs C and U. Then for each wW there is a path γw (unique up to homotopy) joining u0 and w-1(u0). So the choice of C implies a choice of representatives γw in the fundamental group π1(U/W,u0). Here U is the complement of the union of 'bad hyperplanes' (3.13).

We pick a solution Φ of the AKZ equation in U and define the monodromy function 𝒯w (wW) wΦ=w-1 Φ·𝒯ww=w bW. (3.34) Here Φ is invertible at least at one point and is extended analytically to the whole U. The values are in the endomorphisms of any finite-dimensional representation of Σ (we will apply the construction to the induced representations).

The monodromy {𝒯w}wW satisfies the following:

(a) (1-cocycle condition) v-1(𝒯w)𝒯v=𝒯wvw,vW,
(b) ui𝒯w=0, and hence 𝒯w is locally constant.
The property (b) holds since both Φ and w(Φ)=wwΦ satisfy the same differential equation of the first order (the AKZ equation). It readily results in the invertibility of 𝒯w on n-wWw(C). The latter set is not connected, so 𝒯 is not just a constant.

Next let us introduce the operators σw,σw (wW), acting on functions F on U: (σwF)(u) = (w-1F) (u)=F(w(u)) ,σi=σsi, (σwF)(u) = (w-1F) (u)𝒯w, σi=σsi. The relations for the operators σw are the same as for the permutations σw: a)σw σv= σvw, b)σw ub=uw-1(b) σw, c)σwb= w-1(b) σw,b (uα)=(b,α). (3.35) Note that the property a) follows from the 1-cocycle condition for {𝒯w}wW. Indeed, (σwσv) (F) = σw (σv(F)) = σw(v-1F𝒯v) = w-1 (v-1F𝒯v) 𝒯w = w-1v-1F (w-1𝒯v) 𝒯w = (vw)-1 F𝒯vw = σvw(F).

Let 𝒮olAKZ be the space of solutions of the AKZ equation with values in Σ. When we consider the AKZ equation on a finite-dimensional Σ-module V, we will denote the space of its solutions by 𝒮olAKZ(V). Starting with AKZ let us go to QMBP. In what follows, Φ𝒮olAKZ or Φ𝒮olAKZ(End(V)). In the latter case all operators act on End(V)-valued functions.

(1) Using sαΦ=σsαΦ, we rewrite the AKZ equation: xiΦ = ( ui-k αΣ+ ναi sαeuα-1 ) Φ = ( ui-k αΣ+ ναi (euα-1)-1 σsα ) Φ(1in). Let us denote: 𝒟i=ui -kαΣ+ ναi (euα-1)-1 σsα. The local invertibility of Φ and the relations 𝒟iΦ=xiΦ result in the commutativity [𝒟i,𝒟j] =0i,j. Here one can use that the commutators do not contain the derivatives, which readily results from the relations for σ. Moreover, the commutativity follows from these relations algebraically. It was proved in [Che1992] (see [Che1994-3] for a more conceptual proof based on the induced representations). It also follows from the corresponding difference theory, where this and similar statements are much simpler (and completely conceptual).
(2) Since the multiplication by xi commutes with 𝒟j, we get p(x1,,xn) Φ=p(𝒟1,,𝒟n) Φ for any polynomial p[x1,,xn].
(3) For λ=(λ1,,λn)n let us take an Σ-module Vλ with the following properties: (i) p(x1,,xn)= p(λ1,,λn) onVλfor any p[x1,,xn]W, (3.36) (ii) there exists a linear maptr:Vλ satisfying tr(wa)=tr(a) wW,aVλ. (3.37)

Let p(x1,,xn) be a polynomial. Using the commutation relations (3.35), we can write p(𝒟1,,𝒟n) =wW Dw(p) σw, where 𝒟w(p) are differential operators (they do not contain σ). They are scalar and commute with Σ. Thus p(x1,,xn)Φ =wW𝒟w(p) σwΦ=wW 𝒟w(p)wΦ. Now, we assume that p is W-invariant. Applying tr (see (3.36) and (3.37)), we come to p(λ1,,λn) ψ=Lpψfor Lp=wW 𝒟w(p), where ψ(u)=tr(Φ(u)) is a -valued function. The differential operators Lp are W-invariant, which follows from the same construction (we will reprove this algebraically below).

Let us introduce the trigonometric Dunkl operators 𝒟i (1in) replacing σ by σ: 𝒟i=ui-k αΣ+ ναi(euα-1)-1 σsα. Repeating the above construction, define 𝒟(p) for a W-invariant polynomial p by p(𝒟1,,𝒟n)= wW𝒟w(p) σw. Since in the construction of Lp and Lp we use only the commutation relations (3.35) for σw and σi these operators just coincide. The trigonometric Dunkl operators are from [Che1991-2]. Dunkl introduced their rational counterparts (see also [Che1994-3] and references therein). When defining my operators I also used [Hec1991]. Heckman's 'global Dunkl operators' are sufficient to introduce QMBP, but do not commute.

We are now in a position to introduce the QMBP with the eigenvalue λ=(λ1,,λn)n. It is the following system of differential equations for a -valued function ψ. Lpψ=p(λ1,,λn) ψ(p[x1,,xn]W). It is known [HOp1987] (and easy to see by looking at the leading terms of Lp) that the dimension of the space of solutions ψ is |W|.

Summarizing, we come to the theorem.

Theorem 3.8. Applying tr we get a homomorphism tr:𝒮olAKZ (Vλ) 𝒮olQMBP (λ). Here 𝒮olQMBP(λ) denotes the space of solutions to QMBP with the eigenvalue λ.

We can say more for concrete represenatations, especially for the induced representations Jλ (see (3.4)). We define the 'trace' tr:Jλ as the map dual to the embedding Jλ= [x1,,xn] /Lλ sending 1 to 1[x1,,xn]. Here Lλ denotes the ideal generated by p(x)-p(λ), p[x1,,xn]W. One easily checks that tr satisfies the conditions (3.37).

Theorem 3.9 ([Che1991-4]) For any λn, tr gives an isomorphism tr:𝒮olAKZ (Jλ) 𝒮olQMBP(λ).


The key observation: for anyΣ -submodule0M Jλ, we havetr |M0. (3.38) Indeed, if 0fM, then there exists a polynomial p(x)[x1,,xn] such that f(p)0. However f(p)=tr(p(f))tr(M).

To prove Theorem 3.9, it is enough to show the injectivity of tr, since the surjectivity will then follow by comparing the dimensions of the solution spaces (both of them are |W|). So let us suppose that for φ(u)𝒮olAKZ(Jλ) identically tr(φ)=0. (3.39) We will show that tr(Σφ) =0. (3.40) Differentiating (3.39), 0=tr(φui) =kαΣ+ ναitr(sαeuα-1φ) +tr(xiφ). By the W-invariance of tr, tr(sαφ)=tr(φ)=0. Hence tr(xiφ)=0. (3.41) Differentiating this equation by uj we have 0=kαΣ+ ναjtr (xisαeuα-1φ) +tr(xixjφ). Using the commutation relations of xj and sα, we deduce from (3.39), (3.41) that tr(xixjφ)=0. Proceeding in the same way, we establish that tr(xi1xilφ)=0 for any i1,,il. Combining this with the W-invariance of tr, we get (3.40).

For each u0, consider the submodule M=Σφ(u0)Jλ. Then tr|M=0, and from the key observation above, we deduce that M=0. This completes the proof of Theorem 3.9.

The map from Theorem 3.9 was found by Matsuo [Mat1992] for induced representations Iλ. He proved his theorem algebraically (without the passage through the trigonometric Dunkl operators discussed above) using an explicit presentation for AKZ in Iλ. The isomorphism for Jλ (or for Iλ with properly ordered λ - (3.6)) was established independently and simultaneously by Matsuo and the author in [Che1994-3]. He proved that a certain determinant is non-zero for properly ordered λ. I used the modules J. Matsuo was the first to conjecture that the QMBP (the Heckman-Opdam system) and a certain specialization of the trigonometric KZ from [Che1989] are isomorphic. The affine KZ were defined in full generality a bit later (in [Che1991-3]).

Let us give the formula for the simplest Lp.

Example 3.1. Let p2(x1,,xn)=i=1nxαixi. Then we have L2=Lp2= i=1nαi i+αΣ+ (α,α) k(1-k)(euα-e-uα)2 +const. It was studied in [OPe1983].

Remark 3.4. More generally, let A be a [W]-module and VA,λ= (Ind[W]Σ(A)/Lλ). As before, Lλ is the ideal generated by p(x)-p(λ), p[x1,,xn]W. Then the following holds 𝒮olAKZ(VA,λ) 𝒮olQMBPA (λ) where now the right hand side means a matrix version of QMBP (sometimes it is called spin-QMBP). It was introduced in [Che1994-3] for the first time. It is a ceratin unification of the Haldane-Shastry model and that by Calogero-Sutherland.

For example, the L-operator corresponding to p2 above reads L2=i=1n αii+ αΣ+ (α,α) k(sα*-k) (euα-e-uα)2 +const, where by sα* we mean the image of sα in Aut(A).

The GLn case

Let us describe AKZ and QMBP in the GLn case.

In §2.4, we introduced the degenerate affine Hecke algebra of type GLn. It is the algebra n= 𝕊n,y1,,yn subject to the following relations: siyi-yi+1 si=k,si yj=yjsi (ij,j+1), yiyj=yjyi (1i,jn). As in §2.4, we will use the coordinates vi.

To prepare the passage to the difference case, we conjugate the AKZ for GLn by the function Δk for A Δ=i<j(evi-evj). The equation becomes as follows: Φvi= ( k ( j(>i) sij-1evi-vj-1- j(<i) sij-1evj-vi-1 ) +yi+k(i-n+12) ) Φ. (3.42) Only in this form it can be quantized (see §4.2). The system is consistent and 𝕊n-invariant.

The corresponding Dunkl operators are given by the formula 𝒟i=vi-k ( j(>i) (evi-vj-1)-1 (σij-1)- j(<i) (evj-vi-1)-1 (σij-1)+i- n+12 ) . Here σij stands for the transpositions of the coordinates: σijvi=vj σij. Similarly, σw means the permutation of the coordinates corresponding to w-1.

The main point of the theory is that they satisfy the relations from the degenerate Hecke algebra: [𝒟i,𝒟j]=0= [𝒟i,yj], ij,σii+1 𝒟i-𝒟i+1σii+1 =k. It holds for any root systems. This statement is from [Che1994-3]. In these notes we will deduce these relations from the difference theory (where they are almost obvious). These relations readily give that p(𝒟1,,𝒟n) and the corresponding Lp are W-invariant for the W-invariant polynomials. Use the description of the center of to see this.

In the case of GLn, given symmetric p[x1,,xn]𝕊n, p(𝒟1,,𝒟n)= w𝕊n Dw(p)σw, where Dw(p) are scalar differential operators, Lp=p(𝒟1,,𝒟n) |symm.poly.=w𝕊n Dw(p). Let us take the elementary symmetric polynomials: ew(x)= i1<<im xi1xim, as p, setting Lm=Lem. Clearly L1=i=1n vi. The next operator is: L2=i<j 2vivj -k2i<jcoth (vi-vj2) (vi-vj) +k24.

When we replace e2 by p=ixi2, the corresponding L-operator is conjugated (by Δk) to the original Sutherland operator up to a constant term [Sut1971]. For special values of the parameter k, these operators are the radial parts of the Laplace operators on the symmetric spaces. A particular case was considered by Koornwinder. The rational counterpart is due to Calogero. It is equivalent to a rational variant of the AKZ (an extension of the rational W-valued KZ from [Che1989] by the x-s). Here the Jλ-modules cannot be represented as Iλ, the theorem holds in terms of J only (see [Che1994-3]).

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