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

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 $q\text{-deformations.}$ 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 ${ℋ}_{n}^{t}$ in the case of ${GL}_{n}\text{.}$

Let $t\in {ℂ}^{*}$ be a parameter. Then ${ℋ}_{n}^{t}$ is the algebra defined over $ℂ$ by the following set of generators and relations: $generators : T1,…,Tn-1, Y1,…,Yn, relations : (Ti-t) (Ti+t-1) =0,(1≤i≤n-1) (4.1) TiTi+1Ti= Ti+1TiTi+1, (1≤i≤n-2) (4.2) TiTj=TjTi, (|i-j|>1) (4.3) YiYj=YjYi, (1≤i,j≤n) (4.4) YiTj=TjYi, (j≠i,i-1), (4.5) Ti-1Yi Ti-1= Yi+1(1≤i≤n-1). (4.6)$ 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 $P=T1⋯Ti-1 YiTi-1⋯ Tn-1-1.$ It follows from the defining relations (4.1)-(4.6) that the right hand side is independent of $i$ $\text{(}1\le i\le n\text{)}$ and therefore equals to $P=T1⋯Tn-1 Yn=Y1T1-1 ⋯Tn-1-1. (4.7)$

Lemma 4.1. The algebra ${ℋ}_{n}^{t}$ can be presented as $ℋnt= ⟨T1,…,Tn-1,P⟩ /∼, (4.8)$ where the quotient is by the quadratic relations (4.1), the Coxeter relations (4.2)-(4.3) and the following:

 (a) $P{T}_{i-1}={T}_{i}P$ $\text{(}1 (b) ${P}^{n}$ is central. Proof. Notice that in terms of ${Y}_{i}\text{'s}$ we have ${P}^{n}={Y}_{1}\cdots {Y}_{n}\text{.}$ 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 ${T}_{1},\dots ,{T}_{n-1},P$ and introduce the elements ${Y}_{1},\dots ,{Y}_{n}$ by $Y1=PTn-1⋯T1, Y2=T1-1Y1 T1-1,….$ We must check the commutativity ${Y}_{1}{Y}_{2}={Y}_{2}{Y}_{1},$ ${T}_{j}{Y}_{1}={Y}_{1}{T}_{j}$ $\text{(}j>1\text{),}$ etc. using (a), (b). The first reads $Y1T1-1 Y1T1-1= T1-1Y1 T1-1Y1.$ We plug in the above formula for ${Y}_{1}$ 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. $\square$

### The QAKZ equation

In this section, we introduce the QAKZ equation.

Definition 4.1. For $u\in ℂ,$ 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 $\mathrm{\Phi }\left(v\right)$ that takes values in ${ℋ}_{n}^{t}$ (or any ${ℋ}_{n}^{t}\text{-module).}$ $Φ(v1,…,vi+h,…,vn) = Fi-1 (vi-vi+1+h) …F1(vi-v1+h) T1⋯Ti-1Yi ×Ti-1⋯ Tn-1-1 Fn-1 (vi-vn)⋯ Fi(vi-vi+1) ×Φ (v1,…,vi,…,vn) (i=1,…,n). (4.12)$ 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 $\mathrm{\Phi }\left(v\right)$ 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 $h\to 0\text{.}$ The generators ${T}_{i},{Y}_{i}$ are supposed to have the form $Ti = si+kh2+⋯ (si2=1), Yi = 1+hyi+⋯,$ where by $\cdots$ we mean terms of order ${h}^{2}\text{.}$ The relations of the degenerate affine Hecke algebra for ${s}_{i},{y}_{i}$ 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( Hence the AKZ equation (3.42) is a semi-classical limit $\text{(}h\to 0\text{)}$ 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 ${GL}_{n}$ is the semi-direct product $𝕊∼n=𝕊n⋉ ℤn,$ where $ℤn=⨁i=1n ℤγi$ is a free abelian group of rank $n\text{.}$ Define the action of ${\stackrel{\sim }{𝕊}}_{n}$ on a vector $v=\left({v}_{1},\dots ,{v}_{n}\right)\in {ℝ}^{n}$ by $sijv= (v1,…,vj,…,vi,…,vn) =sjiv,i We also introduce $π=γ1s1⋯ sn-1=s1⋯ sn-1γn.$ Its action on ${ℝ}^{n}$ and the coordinates reads as $πv=(vn+h,v1,…,vn-1), πvn=v1-h, πv1=v2,⋯.$

Lemma 4.3. ${\stackrel{\sim }{𝕊}}_{n}$ can be presented as $𝕊∼n= ⟨s1,…,sn-1,π⟩/∼,$ where the relations are $si2=1,si sj=sjsi (|i-j|>1), sisi+1si =si+1sisi+1,$ and

 (a) $\pi {s}_{i-1}{\pi }^{-1}={s}_{i}$ $\text{(}1 (b) ${\pi }^{n}$ is central.

It is convenient to represent the elements ${\gamma }_{i},\pi$ graphically.

Fig.7 shows a reduced decomposition of ${\gamma }_{i}\text{:}$ $γi=si-1⋯ s1πsn-1⋯ si.$

For a function $\mathrm{\Psi }\left(v\right)$ with values in ${ℋ}_{n}^{t},$ let $s∼i(Ψ) = Fi(vi+1-vi) siΨ, (4.13) π∼(Ψ) = PπΨ. (4.14)$

Theorem 4.4 ([Che1992-2]). The formulas (4.13), (4.14) can be extended to an action of ${\stackrel{\sim }{𝕊}}_{n}\text{.}$

$⋮ ⋮ {v}_{i}+h {\gamma }_{i} {v}_{n} {v}_{i+1} {v}_{i} {v}_{i-1} {v}_{1} ⋮ ⋮ \pi {v}_{n}+h {v}_{1} {v}_{n-1} {v}_{n} {v}_{n-1} {v}_{1} Figure. 7. Graphs for γ and π$

We denote this action by ${\stackrel{\sim }{𝕊}}_{n}\ni w:\mathrm{\Psi }↦\stackrel{\sim }{w}\left(\mathrm{\Psi }\right)\text{.}$ For instance, $γ∼i(Ψ) (v1,…,vn)= Fi-1 (vi-1-vi)-1 ⋯F1(v1-vi)-1 PFn-1 (vi-vn-h)⋯ Fi(vi-vi+1-h) ×Ψ(v1,…,vi-h,…,vn).$ Hence the QAKZ equation simply means the invariance of $\mathrm{\Phi }\left(v\right)$ with respect to the pairwise commuting elements ${\gamma }_{i}\text{:}$ $QAKZ⟺γ∼i(Φ) =Φ(i=1,…,n). (4.15)$

Let us connect QAKZ with the q-KZ introduced by Smirnov and Frenkel-Reshetikhin [Smi1986, FRe1991]. We fix an $N\text{-dimensional}$ complex vector space $V$ and introduce $T\in \text{End}\left(V\otimes V\right)$ by $T=(t-t-1) ∑i due to Baxter and Jimbo. The algebra ${ℋ}_{n}^{t}$ acts on ${V}^{\otimes n}$ by $Ti(a1⊗⋯⊗an) = a1⊗⋯⊗T (ai⊗ai+1) ⊗⋯⊗an, (4.16) Pi(a1⊗⋯⊗an) = Can⊗a1⊗⋯⊗ an-1, (4.17)$ where ${a}_{i}\in V$ and $C=\text{diag}\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)\text{.}$ One can check that this action is well-defined by a direct calculation.

For $N=n,$ let $(V⊗n)0=span { ew(1)⊗⋯⊗ ew(n) | w∈𝕊n }$ be the $0\text{-weight}$ subspace. Here ${e}_{1},\dots ,{e}_{n}$ denote the standard basis of $V\text{.}$ It is easy to see that this subspace is closed under the action of ${ℋ}_{n}^{t}\text{.}$ We state the next proposition without proof.

Proposition 4.5. If $N=n$ and $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)$ is generic, then the $0\text{-weight}$ space ${\left({V}^{\otimes n}\right)}_{0}$ is isomorphic to ${I}_{\lambda }={\text{Ind}}_{ℂ\left[{Y}_{1},\dots ,{Y}_{n}\right]}^{{ℋ}_{n}^{t}}\left(\lambda \right)\text{.}$

Writing down AQKZ in ${\left({V}^{\otimes n}\right)}_{0}$ we get the q-KZ (for ${GL}_{n}$ 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 $\mathrm{\Phi }$ be a solution of the QAKZ equation. Thanks to (4.15), $\stackrel{\sim }{w}\left(\mathrm{\Phi }\right)$ is also a solution of the QAKZ equation for any $w\in {\stackrel{\sim }{𝕊}}_{n}\text{.}$ We define ${𝒯}_{w}\in {ℋ}_{n}^{t}$ by $w𝒯w=Φ-1 w∼(Φ)for w∈𝕊∼n$ 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 ${𝒯}_{i}$ stands for ${𝒯}_{{s}_{i}}\text{.}$

Lemma 4.6. $w2-1 (𝒯w1) 𝒯w2=𝒯w1w2 for w1,w2∈ 𝕊∼n.$

Indeed, $Φw1w2 𝒯w1w2= w1w2˜ (Φ)=w∼1 (w∼2Φ)= w∼1 (Φw2𝒯w2) =Φw1𝒯w1 w1w2𝒯w2.$

The QAKZ equation implies that ${𝒯}_{{\gamma }_{i}}=1\text{.}$ Hence ${𝒯}_{w}$ depends only on the image $\stackrel{‾}{w}$ of $w$ in ${𝕊}_{n}\text{.}$

Let $ℱ\left({ℂ}^{n},{ℋ}_{n}^{t}\right)$ be the set of ${ℋ}_{n}^{t}\text{-valued}$ function on ${ℂ}^{n}\text{.}$ Next we define two anti-actions of ${\stackrel{\sim }{𝕊}}_{n}\text{:}$ $σw(Ψ) = w-1Ψ, (4.20) σw′(Ψ) = w-1Ψ 𝒯w, (4.21)$ where $w\in {\stackrel{\sim }{𝕊}}_{n}$ and $\mathrm{\Psi }\in ℱ\left({ℂ}^{n},{ℋ}_{n}^{t}\right)\text{.}$ Lemma 4.6 means exactly that $\sigma \prime$ is an anti-action ${\sigma }_{{w}_{1}{w}_{2}}^{\prime }={\sigma }_{{w}_{2}}^{\prime }{\sigma }_{{w}_{1}}^{\prime }\text{).}$ For instance, ${\sigma }_{{\gamma }_{i}}\left({v}_{i}\right)={v}_{i}+h={\sigma }_{{\gamma }_{i}}^{\prime }\left({v}_{i}\right)\text{.}$

We note that in the difference theory the monodromy can be always made trivial. Indeed, the 1-cocycle $\left\{{𝒯}_{w},w\in W\right\}$ 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 $\sigma ,\sigma \prime$ are not too much different in contrast to the differential theory.

This argument can be applied to the AQKZ itself, although the group ${ℤ}^{n}$ is infinite. We can formally solve the QAKZ equation as follows. Let $\mathrm{\Psi }\in ℱ\left({ℂ}^{n},{ℋ}_{n}^{t}\right)\text{.}$ Then the infinite sum $∑b∈Bb∼ (Ψ), (4.22)$ where $B=\underset{i=1}{\overset{n}{\oplus }}ℤ{\gamma }_{i}\subset {\stackrel{\sim }{𝕊}}_{n},$ satisfies the AQKZ, provided the convergence. For example, if $\mathrm{\Psi }$ is rapidly decreasing, then one can check that $\sum _{b\in B}\stackrel{\sim }{b}\left(\mathrm{\Psi }\right)$ is convergent.

We see, that constructing $\text{End}\left(V\right)\text{-valued}$ solution $\mathrm{\Phi }$ to QAKZ for finite-dimensional ${ℋ}_{n}^{t}\text{-modules}$ $V$ 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 $q\text{-analogue}$ of the relation between AKZ and QMBP discussed in §3.4.

Let $\mathrm{\Phi }$ be a solution of the QAKZ equation with values in $\text{End}\left(V\right)$ for a ${ℋ}_{n}^{t}\text{-module}$ $V\text{.}$ We assume that it is invertible for sufficiently general $v\text{.}$ Setting ${\sigma }_{i}^{\prime }={\sigma }_{{s}_{i}}^{\prime },$ 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 ${\stackrel{ˆ}{T}}_{i}^{\prime }$ $\text{(}1\le i\le n\text{)}$ by $Tˆi′=t σi′+ t-t-1 evi-vi+1-1 (σi′-1). (4.23)$ Then ${\stackrel{ˆ}{T}}_{i}^{\prime }\mathrm{\Phi }={T}_{i}\mathrm{\Phi }$ and ${\sigma }_{\pi }^{\prime }\mathrm{\Phi }=P\mathrm{\Phi }$ (see (4.19)). The operators ${\stackrel{ˆ}{T}}_{i}^{\prime }$ and ${\sigma }_{\pi }^{\prime }$ commute with the left multiplication by ${T}_{j},$ $P$ and any elements from ${ℋ}_{n}^{t}\text{.}$ Using all these: $YiΦ = Ti-1-1⋯ T1-1P Tn-1⋯Ti+1 TiΦ = Ti-1-1⋯ T1-1P Tn-1⋯Ti+1 Tˆi′Φ = Tˆi′Ti-1-1⋯ T1-1P Tn-1⋯Ti+1 Φ ⋯ = Tˆi′⋯ Tˆn-1′ σπ′ (Tˆ1′)-1 ⋯(Tˆi-1′)-1 Φ.$ We come to the following definition: $Δi′= Tˆi′⋯ Tˆn-1′ σπ′ (Tˆ1′)-1 ⋯(Tˆi-1′)-1 ,1≤i≤n. (4.24)$

Since ${Y}_{i}\mathrm{\Phi }={\mathrm{\Delta }}_{t}^{\prime }\mathrm{\Phi }$ and ${Y}_{i}$ commute with each other, $[Δi′,Δj′] =0.$ By the construction, the operators ${\mathrm{\Delta }}_{i}^{\prime }$ act in $\text{End}\left(V\right)\text{-valued}$ functions. However if we understand them formally the commutativity can be deduced from the relations $σi′vi= vi+1σi′, (4.25) σi′σγi= σγi+1σi′. (4.26) σγi′= σγi (4.27)$ The latter means that ${𝒯}_{{\gamma }_{i}}=1\text{.}$

Let $Q$ be a polynomial in $n$ variables. Then $Q(Y1,…,Yn)Φ =Q(Δ1′,…,Δn′) Φ$ and we can represent $Q(Δ1′,…,Δn′) =∑w∈𝕊n Dw(Q)σw′, (4.28)$ where ${D}_{w}^{\left(Q\right)}$ are pure difference operators, which do not contain ${\sigma }_{w}^{\prime }$ $\text{(}w\in {𝕊}_{n}\text{).}$

For symmetric $Q,$ we introduce a difference operator of Macdonald type ${M}_{Q}$ by $MQ=∑w∈𝕊n Dw(Q).$

Let $\phi$ be a $ℂ\text{-valued}$ function on ${ℂ}^{n}\text{.}$ The system $MQφ=Q (λ1,…,λn)φ (4.29)$ will be called the Macdonald eigenvalue problem. The operators ${M}_{Q}$ can be calculated for $\sigma$ instead of $\sigma \prime \text{.}$ As in the differential case, the result will be the same.

Fix $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)\in {ℂ}^{n}\text{.}$ We take a left ${ℋ}_{n}^{t}\text{-module}$ ${V}_{\lambda }$ with the following properties:

 (1) for any symmetric polynomial $Q$ in $n$ variables and all $a\in {V}_{\lambda },$ $Q(Y1,…,Yn)a= Q(λ1,…,λn)a,$ (2) there exists a $ℂ\text{-linear}$ map $\text{tr}:{V}_{\lambda }\to ℂ$ such that $tr((Ti-t)a)=0$ for all $i$ and $a\in {V}_{\lambda }\text{.}$

As always, we fix a (local) invertible solution $\mathrm{\Phi }\left(v\right)$ of the QAKZ equation with values in $\text{End}\left({V}_{\lambda }\right)\text{.}$ Note that all ${V}_{\lambda }\text{-valued}$ solutions of the QAKZ equation can be written in the form $\phi \left(v\right)=\mathrm{\Phi }\left(v\right)a\left(v\right)$ for $B\text{-periodic}$ ${V}_{\lambda }\text{-valued}$ function $a\left(v\right)\text{:}$ $a(…,vi+h,…) =a(v) for i=1, …,n.$

Theorem 4.7. Let ${V}_{\lambda }$ be an ${ℋ}_{n}^{t}\text{-module}$ with the above properties, ${𝒮\text{ol}}_{\text{QAKZ}}\left({V}_{\lambda }\right)$ be the space of solutions of the QAKZ equation with values in ${V}_{\lambda },$ and ${𝒮\text{ol}}_{\text{Mac}}\left(\lambda \right)$ the space of solutions of the Macdonald eigenvalue problem (4.29). Then $𝒮olQAKZ(Vλ) →tr𝒮olMac(λ).$ Proof. Let $\phi \left(v\right)=\mathrm{\Phi }\left(v\right)a\in {𝒮\text{ol}}_{\text{QAKZ}}\left({V}_{\lambda }\right)\text{.}$ Then $(σi′-1)Φ= (t+t-t-1evi-vi+1-1)-1 (Ti-t)Φ.$ For a reduced decomposition $w={s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ of $w\in {𝕊}_{n},$ $σw′-1 = σil′⋯ σi1′-1 = σsil′⋯ σsi2′ (σsi1′-1) +σil′⋯ σi2′-1 ⋯ = ∑k=1l σil′⋯ σik+1′ (σik′-1).$ Since ${\sigma }_{i}^{\prime }$ commutes with the left action of $\left\{T\right\},$ 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 ${D}_{w}^{\left(Q\right)}$ with ${T}_{i}-t$ we represent ${D}_{w}^{\left(Q\right)}\left({\sigma }_{w}^{\prime }-1\right)\mathrm{\Phi }$ as a sum $\sum \left({T}_{i}-t\right){\mathrm{\Psi }}_{i}$ for some ${ℋ}_{n}^{t}\text{-valued}$ functions ${\mathrm{\Psi }}_{i}\text{.}$ Finally $Q(λ1,…,λn) Φ = Q(Δ1′,…,Δn′) Φ = ∑w∈𝕊n Dw(Q) σw′Φ = ∑w∈𝕊n Dw(Q)Φ +∑w∈𝕊n Dw(Q) (σw′-1)Φ = MQΦ+∑ (Ti-t)Ψi.$ Applying this relation to $a\in {V}_{\lambda }$ and taking tr, we conclude: $Q(λ1,…,λn) tr(φ)=MQtr(φ).$ $\square$

Let us now consider ${V}_{\lambda }={J}_{\lambda }^{\circ }\text{.}$ The definition is quite similar to the differential case. We start with $Jλ=IndHntℋnt (+)/Lλ.$ Here ${H}_{n}^{t}=⟨{T}_{1},\dots ,{T}_{n-1}⟩\subset {ℋ}_{n}^{t},$ $+:{H}_{n}^{t}⟶ℂ$ is the one-dimensional representation sending ${T}_{i}$ to $t,$ and ${L}_{\lambda }$ is the ideal generated by $p\left({Y}_{1},\dots ,{Y}_{n}\right)-p\left(\lambda \right)$ $\text{(}p\in ℂ{\left[{x}_{1},\dots ,{x}_{n}\right]}^{{𝕊}_{n}}\text{).}$ As in §3.1, ${J}_{\lambda }^{\circ }$ stands for the dual module defined via the anti-involution $\circ$ of ${ℋ}_{n}^{t}\text{:}$ $Yi∘=Yi, Ti∘=Ti.$

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

Theorem 4.8. If ${V}_{\lambda }={J}_{\lambda }^{\circ },$ then the map from ${𝒮\text{ol}}_{\text{QAKZ}}\left({V}_{\lambda }\right)$ to ${𝒮\text{ol}}_{\text{Mac}}\left(\lambda \right)$ is injective.

The theorem results from the following two lemmas.

Lemma 4.9. Let $K$ be a ${ℋ}_{n}^{t}\text{-submodule}$ of ${J}_{\lambda }^{\circ }\text{.}$ Then $\text{tr}\left(K\right)=0$ implies $K=0\text{.}$

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

Lemma 4.10. Let $\phi$ be a ${V}_{\lambda }\text{-valued}$ solution of the QAKZ equation. Assume $\text{tr}\left(\phi \right)=0\text{.}$ Then $\text{tr}\left({ℋ}_{n}^{t}\phi \right)=0\text{.}$ Proof.

First $tr(Tiφ)=t tr(φ)=0$ for all $i\text{.}$ Then $\pi ={s}_{1}{s}_{2}\cdots {s}_{n-1}{\gamma }_{n},$ $σπ′-1 = σγn σn-1′⋯ σ1′-1 = σγn ( ∑k=1n-1 σn-1′⋯ σk+1′ (σk′-1) ) +σγn-1,$ and $\text{tr}\left({\sigma }_{{\gamma }_{n}}\phi \right)=\text{tr}\left(\phi \right)=0\text{.}$ Therefore, representing $\phi =\mathrm{\Phi }a$ $\text{(}a\in {V}_{\lambda }\text{),}$ we have $tr(Pφ)-tr(φ) = tr((σπ′-1)Φa) = tr ( σγn ( ∑k=1n-1 σn-1′⋯ σk+1′ (σk′-1)Φ ) a ) = ∑k=1n-1tr ( σγnσn-1′ ⋯σk+1′fi (v)(Tk-t)Φa ) = ∑k=1n-1tr ( (Tk-t) σγnσn-1′ ⋯σk+1′fi (v)Φa ) = 0,$ where ${f}_{i}\left(v\right)$ are $ℂ\text{-valued}$ function. Hence $\text{tr}\left(P\phi \right)=0$ and $tr(Ynφ) = tr(Tn-1-1⋯T1-1Pφ) = t1-ntr(Pφ) = 0.$

Now we shall prove that $\text{tr}\left({Y}_{i}\phi \right)=0$ for all $i$ by induction. Assume that $\text{tr}\left({Y}_{i}\phi \right)=0$ for $k+1\le i\le n\text{.}$ Since ${Y}_{k}={T}_{k-1}^{-1}\cdots {T}_{1}^{-1}P{T}_{n-1}\cdots {T}_{k}$ it is enough to see that $\text{tr}\left(P{T}_{n-1}\cdots {T}_{k}\phi \right)=0\text{.}$ Since $\phi$ 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 ${c}_{i}\left(v\right)$ and ${f}_{i}\left(v\right)$ are some scalar functions. Therefore $0=tr (PFn-1⋯Fkφ) = tr ( cn-1⋯ckP (Tn-1+fn-1(v)) ⋯(Tk+fk(v)) φ ) = ∑I=(i1,…,il) tr(cI(v)PTil⋯Ti1φ)$ where $I=\left({i}_{1},\dots ,{i}_{l}\right)$ is a sequence of integers such that $k\le {i}_{1}<{i}_{2}<\cdots <{i}_{l}\le n-1,$ and ${c}_{I}\left(v\right)$ is some scalar function. If $I\ne {I}_{0}=\left(k,k+1,\dots ,n-1\right)$ then there are the following possibilities:

 (1) ${i}_{l}\ne n-1,$ (2) ${i}_{l}=n-1$ and there exists an $m$ $\text{(}1\le m\le l\text{)}$ such that ${i}_{j}-{i}_{j-1}=1$ for any $j=m+1,m+2,\dots ,l$ and ${i}_{m}-{i}_{m-1}>1,$ (3) otherwise.

case (1): As ${i}_{l} we have $tr(PTil⋯Ti1φ) = tr(Til+1⋯Til+1Pφ) = tltr(Pφ) = 0.$

case (2): Since $\left[{T}_{i},{T}_{j}\right]=0$ for $|i-j|>1,$ $tr ( P(Til⋯Tim) (Tim-1⋯Ti1) φ ) = tr ( P(Tim-1⋯Ti1) (Til⋯Tim)φ ) = tr (Tim-1+1⋯Ti1+1 PTil⋯Timφ ) = tm-1tr (PTil⋯Timφ).$ By the induction hypothesis, $\text{tr}\left(P{T}_{{i}_{l}}\cdots {T}_{{i}_{m}}\phi \right)=0\text{.}$ Hence $tr(PTil⋯Ti1φ)=0.$

case (3): In this case $I=\left({i}_{1},\dots ,{i}_{l}\right)$ must be of the form ${i}_{l}=n-1,$ ${i}_{l-1}=n-2,$ $\dots ,$ ${i}_{1}=n-l>k\text{.}$ By induction, $\text{tr}\left(P{T}_{{i}_{l}}\cdots {T}_{{i}_{1}}\phi \right)=0\text{.}$ So $\text{tr}\left({Y}_{i}\phi \right)=0$ for all $i\text{.}$

Because of the relations between $T$ and $Y,$ it remains to check that $\text{tr}\left({Y}_{{i}_{1}}\cdots {Y}_{{i}_{l}}\phi \right)=0$ for any $l\text{.}$ One can show this by induction on $l\text{.}$

$\square$

### Macdonald operators

We set $Tˆi = tσi+ t-t-1evi-vi+1-1 (σi-1), (1≤i≤n-1), (4.30) Gij = t+ t-t-1evi-vj-1 (1-σij), (1≤i,j≤n), (4.31) Δi = Tˆi⋯ Tˆn-1σπ Tˆ1-1⋯ Tˆi-1-1. (4.32)$ Here ${\sigma }_{w}$ are from (4.20), ${\sigma }_{ij}={\sigma }_{{s}_{ij}}\text{.}$

Switching from $\left\{T\right\}$ to $\left\{G\right\}\text{:}$ $Tˆiσi = Gi i+1, Gij-1 = t-1- t-t-1evi-vj-1 (1-σij), Δi = Gi i+1⋯ Gi nσγi G1 i-1⋯ Gi-1 i-1.$ Let ${e}_{m}$ be the $m\text{-th}$ elementary symmetric polynomial in $n$ variables. We represent $em(Δ1,…,Δn)= ∑w∈𝕊nDw(m) σw, (4.33)$ for difference operators ${D}_{w}^{\left(m\right)},$ and define $Mm=Mem= ∑w∈𝕊n Dw(m).$ All these operators are $W\text{-invariant,}$ which results from the following lemmas.

Lemma 4.11. Consider the algebra $\stackrel{ˆ}{ℋ}$ generated by ${\stackrel{ˆ}{T}}_{i}$ $\text{(}1\le i\le n-1\text{),}$ ${\mathrm{\Delta }}_{j}$ $\text{(}1\le j\le n\text{).}$ Then ${T}_{i}↦{\stackrel{ˆ}{T}}_{i},$ ${Y}_{j}↦{\mathrm{\Delta }}_{j}$ extends to an algebra isomorphism ${ℋ}_{n}^{t}\stackrel{\sim }{\to }\stackrel{ˆ}{ℋ}\text{.}$ Moreover, if $Q$ is a symmetric polynomial in $n$ variables, then $Q\left({\mathrm{\Delta }}_{1},\dots ,{\mathrm{\Delta }}_{n}\right)$ is a central element in $\stackrel{ˆ}{ℋ}\text{.}$

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 $\stackrel{ˆ}{ℋ}$ with ${ℋ}_{n}^{t}\text{.}$

Lemma 4.12. Let $f\left({v}_{1},\dots ,{v}_{n}\right)$ be a function on ${ℂ}^{n}\text{.}$ Then $f$ is symmetric if and only if $\left({\stackrel{ˆ}{T}}_{i}-t\right)f=0$ for all $i\text{.}$

Lemma 4.13. Let $Q$ be a symmetric polynomial in $n$ variables. Then $Q\left({\mathrm{\Delta }}_{1},\dots ,{\mathrm{\Delta }}_{n}\right)$ acts on the space of the symmetric polynomials in ${e}^{{v}_{i}}$ $\text{(}1\le i\le n\text{).}$ Proof. This follows immediately from Lemma 4.11 and 4.12. $\square$

Let us calculate ${M}_{1}\text{.}$ Since ${M}_{1}$ is symmetric, it is enough to find the coefficient of ${\sigma }_{{\gamma }_{1}}\text{.}$ Using the $G\text{-representation}$ it is easy to see that ${\sigma }_{{\gamma }_{1}}$ does not appear in ${\mathrm{\Delta }}_{2}\text{.}\dots ,{\mathrm{\Delta }}_{n}\text{.}$ The ${\sigma }_{{\gamma }_{1}}\text{-factor}$ of ${\mathrm{\Delta }}_{1}$ is equal to $\prod _{i=2}^{n}\frac{t{e}^{{v}_{1}-{v}_{i}}-{t}^{-1}}{{e}^{{v}_{1}-{v}_{i}}-1}{\sigma }_{{\gamma }_{1}}\text{.}$

After the symmetrization we get the formula: $M1=∑i=1n ∏j≠i tevi-t-1evj evi-evj σγi.$

Similarly, $Mm=∑I=(i1,…,im) ∏i∈I,j∉I tevi-t-1evj evi-evj σγi1⋯ σγim$ where $I=\left({i}_{1},\dots ,{i}_{m}\right)$ is a sequence of integers such that $1\le {i}_{1}<\cdots <{i}_{m}

To recapitulate, let us consider the classical limit of the Macdonald operators. Setting $q={e}^{h}$ and $t={q}^{k/2},$ $h\to 0,$ we have $Δi = 1+h𝒟i+O(h2), M1-n = h∑∂∂vi+ O(h2), M2-(n-1)M1+ n(n-1)2 = h2 ( L2+(n+13) k3 ) +O(h3).$

Remark 4.1. Take a solution $\mathrm{\Phi }=\mathrm{\Phi }\left(v\right)$ of the QAKZ equation in an ${ℋ}^{t}\text{-module}$ $V,$ assuming that $\mathrm{\Phi }$ has the trivial monodromy. Then, for any polynomial $p\in ℂ\left[{x}_{1},\dots ,{x}_{n}\right],$ we have $p(Y1,…,Yn) Φ=p(Δ1,…,Δn) Φ, (4.34)$ where ${\mathrm{\Delta }}_{i}$ are the difference Dunkl operators defined before. Note that ${\mathrm{\Delta }}_{i}^{\prime }$ can be replaced by ${\mathrm{\Delta }}_{i}$ because the monodromy of $\mathrm{\Phi }$ is trivial. We also need a linear functional $\text{pr}:{V}_{\lambda }\to ℂ$ for a vector $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)\in {ℂ}^{n}$ such that $pr(Yib)=λi pr(b)(i=1,…,n) (4.35)$ for any $b\in V\text{.}$ Given any element $a\in V,$ let us define a scalar-valued function $\phi =\phi \left(v\right)$ setting $φ(v)=pr (Φ(v)a)∈ℂ. (4.36)$ Then the formula (4.34) implies $p(λ1,…,λn) φ=p(Δ1,…,Δn) φ. (4.37)$ Thus, the scalar-valued function $\phi =\phi \left(v\right)$ solves the Dunkl eigenvalue problem.

Remark 4.2. Arbitrary root systems. Let $\mathrm{\Sigma }=\left\{\alpha \right\}\in {ℝ}^{n}$ be any reduced root system of rank $n$ (of type $A,$ $B,$ $C,$ $D,$ $E,$ $F$ or $G\text{),}$ and $ℋt= ⟨ T1,…,Tn, X1,…,Xn ⟩ (4.38)$ the corresponding affine Hecke algebra. The baxterization (a parametric deformation satisfying the Yang-Baxter relations) of ${T}_{i}$ will be given by $Fi=Ti+ t-t-1 eui-1 withui= (u,αi) (4.39)$ for each $i=1,\dots ,n\text{.}$ We also have to use the element $T0=Xθ∨ Tθ-1 (4.40)$ corresponding to the simple affine root ${\alpha }_{0}=\delta -\theta$ for $\theta$ being the highest root. Its baxterization is quite similar: $F0=T0+ t-t-1 eh-uθ-1 , (4.41)$ where ${u}_{\theta }=\left(u,\theta \right)\text{.}$ The functions ${F}_{0},{F}_{1},\dots ,{F}_{n}$ satisfy the Yang-Baxter equations associated with the extended Dynkin diagram. For example, in the case of ${\circ }^{1}⇒{\circ }^{2},$ 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 ${F}_{i}$ can also be determined graphically by means of the equivalent pictures of the reflection of two particles (see [Che1992-2]).

Using ${T}_{0},$ the affine Hecke algebra ${ℋ}^{t}$ has an alternative representation $ℋt= ⟨T0,T1,…,Tn;Π⟩, (4.43)$ where $\mathrm{\Pi }$ is a certain finite abelian group. The group $\mathrm{\Pi }$ is isomorphic to ${P}^{\vee }/{Q}^{\vee }\text{.}$ It is the set of all elements of the extended affine Weyl group $W∼=W⋉B,B= ⨁i=1nℤbi, (4.44)$ preserving the set $\left\{{\alpha }_{0},{\alpha }_{1},\dots ,{\alpha }_{n}\right\}$ of the simple affine roots. It gives the embedding of $\mathrm{\Pi }$ into the automorphism group of the extended Dynkin diagram. The action of $\stackrel{\sim }{W}$ on ${ℝ}^{n}\oplus ℝ\delta$ is by the affine reflections and the corresponding shifts in the $\delta \text{-direction}$ for $B\text{:}$ $b(z+ζδ)=z+ (ζ-(b,z))δ.$

Lemma 4.14. $\stackrel{\sim }{W}=⟨{s}_{0},{s}_{1},\dots ,{s}_{n};\mathrm{\Pi }⟩$ with ${s}_{0}=\left({\theta }^{\vee }\right)·{s}_{\theta }\text{.}$

The group $\mathrm{\Pi }$ can be embedded into the affine Hecke algebra. The images ${P}_{\pi }$ of the elements $\pi \in \mathrm{\Pi }$ permute $\left\{{T}_{i}\right\}$ in the same way as $\pi$ do in $\stackrel{\sim }{W}$ with $\left\{{s}_{i}\right\}\text{.}$ The baxterization of the elements in $\mathrm{\Pi }$ is trivial: ${F}_{\pi }={P}_{\pi }$ for each $\pi \in \mathrm{\Pi }\text{.}$

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

Theorem 4.15. Given any ${ℋ}^{t}\text{-valued}$ function $\mathrm{\Psi }=\mathrm{\Psi }\left(u\right),$ the formulas $s∼i(Ψ)= si(FiΨ) (4.45)$ for all $i=0,1,\dots ,n,$ and $π∼(Ψ)= PππΨ (4.46)$ for all $\pi \in \mathrm{\Pi }$ induce a representation of $\stackrel{\sim }{W}\text{.}$

The QAKZ equation for $\mathrm{\Sigma }$ is the invariance condition $\stackrel{\sim }{b}\left(\mathrm{\Phi }\right)=\mathrm{\Phi }$ for all $b\in B\text{.}$ It can be shown that this equation is equivalent to the difference QMBP associated with the root system $\mathrm{\Sigma }$ 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]).