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
aram@unimelb.edu.au

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 ${\mathscr{H}}_{\mathrm{\Sigma }}^t$ and connect them with the degenerate affine Hecke algebra ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ 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 ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$

In this section we define induced representations of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$

For $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {ℂ}^n,$ the character of $ℂ[x_1,\dots ,x_n]$ (i.e. a ring homomorphism $ℂ[x_1,\dots ,x_n]\in ℂ\text{)}$ is an assignment $x_i\mapsto {\lambda }_i\text{.}$ We denote it by $\lambda \text{.}$

Definition 3.1. We define an ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-module}$ $I_{\lambda }$ as the representation induced from $\lambda \text{:}$ $$\begin{array}{cc}{I}_{\lambda }={\text{Ind}}_{ℂ[x_1,\dots ,x_n]}^{{\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }}\left(\lambda \right)={\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }{\otimes }_{ℂ[x_1,\dots ,x_n]}{ℂ}_{\lambda }\text{.}& \text{(3.1)}\end{array}$$ Here ${ℂ}_{\lambda }$ is endowed with the $ℂ[x_1,\dots ,x_n]\text{-module}$ structure by the character $\lambda \text{.}$

We have the Poincaré-Birkhoff-Witt type theorem for ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ Namely any $h\in {\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ is expressed uniquely in either of the following ways: $$\begin{array}{cc}{\displaystyle h=\sum _{w\in W}{p}_w\left(x\right)w=\sum _{w\in W}w{q}_w\left(x\right)}& \text{(3.2)}\end{array}$$ with $p_w,q_w\in ℂ[x_1,\dots ,x_n]\text{.}$ The existence results from the relations (2.32)-(2.34) in ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ Hence $$\begin{array}{cc}{\displaystyle I_{\lambda }=ℂ\left[W\right]={\oplus }_{w\in W}ℂw\text{.}}& \text{(3.3)}\end{array}$$ Thus $I_{\lambda }$ is $ℂ\left[W\right]$ as a $W\text{-module,}$ where the action of $x_i$ is determined by $x_i\left(e\right)={\lambda }_{i}e$ for the identity $e\in W\text{.}$ The action of $x_i$ on other elements of $ℂ\left[W\right]$ 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 $+:W⟶ℂ,$ $w\to 1\text{.}$ Then $$\begin{array}{cc}J={\text{Ind}}_{ℂ\left[W\right]}^{{\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }}(+),& \text{(3.4)}\end{array}$$ is isomorphic to $ℂ[x_1,\dots ,x_n]$ as a vector space and moreover as a $ℂ[x_1,\dots ,x_n]\text{-module.}$ To get finite-dimensional representations from $J,$ we use the coincidence of the center of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ with the algebra of $W\text{-invariant}$ polynomials in $x_i\text{.}$ This theorem is due to Bernstein. The procedure is as follows. Let us fix an element $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {ℂ}^n$ and introduce the ideal $L_{\lambda }$ in $ℂ[x_1,\dots ,x_n]$ generated by $p\left(x\right)-p\left(\lambda \right)$ for all $W\text{-invariant}$ polynomials $p\text{.}$ Set $J_{\lambda }=J/L_{\lambda }\text{.}$ Then $J_{\lambda }$ has a structure of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-module}$ by virtue of the Bernstein theorem.

We will also use the anti-involution $˚$ on ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{:}$ $$\begin{array}{cc}{x}_i^{\circ }=x_i,s_i^{\circ }=s_i,{\left(ab\right)}^{\circ }=b^{\circ }{a}^{\circ },k^{\circ }=k\text{.}& \text{(3.5)}\end{array}$$ Since the relations of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ are self-dual it is well-defined. For an ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-module}$ $V,$ we consider its dual ${\text{Hom}}_{ℂ}(V,ℂ)\text{.}$ The dual has an anti-action (a right action) of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ Composing it with the anti-automorphism $˚,$ we get a natural (left) action of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ We denote the resulting module by $V^{\circ }\text{.}$

We write ${\lambda }_b=\sum k_i{\lambda }_i$ for $b=\sum k_i{b}_i\text{.}$

Theorem 3.1.

(a)	$I_{\lambda }$ is irreducible if and only if ${\lambda }_{{\alpha }^{\vee }}\ne \pm k$ for any $\alpha \in {\mathrm{\Sigma }}_{+}\text{.}$
(b)	There exists a permutation $\lambda \prime$ of $\lambda$ (i.e. $\lambda \prime =w\left(\lambda \right)$ for $w\in W\text{)}$ such that ${\lambda }_{{\alpha }^{\vee }}^{\prime }\ne -k$ for any $\alpha \in {\mathrm{\Sigma }}_{+}\text{.}$ Then $$\begin{array}{cc}{J}_{\lambda }\simeq I_{\lambda \prime }\text{.}& \text{(3.6)}\end{array}$$
(c)	For the longest element $w_0$ in $W,$ $$\begin{array}{cc}{I}_{\lambda }^{\circ }=I_{w_0\left(\lambda \right)}& \text{(3.7)}\end{array}$$

A key lemma in proving Theorem 3.1 is

Lemma 3.2. $I_{(0,\dots ,0)}$ 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 $1\le i\le n$ we set $$\begin{array}{cc}{\displaystyle f_i=f_{s_i}=s_i-\frac{k}{x_{a_i}}\text{.}}& \text{(3.8)}\end{array}$$ For $w\in W$ with a reduced decomposition $w=s_{i_n}\cdots s_{i_1},$ $f_w=f_{i_n}\cdots f_{i_1}\text{.}$ We call the elements $f_w$ intertwiners.

The elements $f_w$ belong to the localization of the degenerate affine Hecke algebra ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ by the $W\text{-invariant}$ polynomials. They give a certain 'baxterization' of $w,$ and are closely related to the Yang's $R\text{-matrix.}$ Let us show that $f_w$ does not depend on the choice of the reduced decomposition of $w\text{.}$

We have $$\begin{array}{cc}{f}_{s_i}{x}_b=x_{s_i\left(b\right)}{f}_{s_i},& \text{(3.9)}\\ f_w{x}_b=x_{w\left(b\right)}{f}_w\text{.}& \text{(3.10)}\end{array}$$ Indeed, $$\begin{array}{cc}{\displaystyle (s_i-\frac{k}{x_{{\alpha }_i}})x_b=x_{s_i\left(b\right)}(s_i-\frac{k}{x_{{\alpha }_i}}),}& \text{(3.11)}\end{array}$$ which can be rewritten as follows: $$\begin{array}{cc}{\displaystyle s_i{x}_b-x_{s_i\left(b\right)}{s}_i=-k\frac{x_{s_i\left(b\right)}-x_b}{x_{{\alpha }_i}}\text{.}}& \text{(3.12)}\end{array}$$ Using the definition of $x_b$ (2.36), the right hand side of (3.12) is $k(b,{\alpha }_i)\text{.}$ So we come to (2.37). The relations (3.10) fix $f_w$ uniquely up to the multiplication on the right by functions in $x\text{.}$ The leading terms of $f_w$ being $w,$ they coincide for any reduced decompositions.

To demonstrate the role of intertwiners, let us check the irreducibility of $I_{\lambda }$ for generic $\lambda \text{.}$ First note that the vectors $\{f_w\left(e\right)\in I_{\lambda }\}$ are common eigenvectors of $x_b,$ because $$x_b{f}_w\left(e\right)=f_w{x}_{w^{-1}\left(b\right)}\left(e\right)={\lambda }_{w^{-1}\left(b\right)}{f}_w\left(e\right),$$ For a generic $\lambda$ the eigenvalues are simple, hence these vectors are linearly independent. Now, any nonzero ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-submodule}$ $A$ of $I_{\lambda }$ contains at least one eigenvector of $x_b\text{.}$ By the simplicity of eigenvalues, such an eigenvector must be in the form $f_w\left(e\right)$ for some $w\in W\text{.}$ On the other hand $f_w$ are invertible elements. Indeed, $$f_i^{-1}={(1-\frac{k^2}{x_{{\alpha }_i}^2})}^{-1}{f}_i\text{.}$$ Therefore $e\in A\text{.}$ Since $I_{\lambda }$ is generated by $e,$ we conclude that $A=I_{\lambda }\text{.}$

Actually this very reasoning leads to the proof of the Theorem (a),(b). However if $\lambda$ 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 $\lambda \text{.}$

Remark 3.1. The $\mathscr{H}\prime \text{-quotients}$ $A$ of $J_{\lambda }^{\circ }$ will be interpreted below as certain quotients of the $D\text{-module}$ representing the quantum many-body eigenvalue problem. A solution of the AKZ in $J_{\lambda }^{\circ }$ induces solutions in any of its $\mathscr{H}\prime \text{-quotients}$ (if $I$ is reducible). It gives a one-to-one correspondence between the $\mathscr{H}\prime \text{-submodules}$ (quotients, constituents) of $J$ and those of the $D\text{-modules}$ 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 $J_{\lambda }^{\circ }$ and the quantum many-body problem (QMBP) with the eigenvalue $\lambda \text{.}$

Let $U\prime$ be the open subset of ${ℂ}^n$ given by $$\begin{array}{cc}{\displaystyle U\prime =\{u\in {ℂ}^n\hspace{0.17em}|\hspace{0.17em}\prod _{\alpha \in {\mathrm{\Sigma }}_{+}}(e^{u_{\alpha }}-1)\ne 0\}\text{.}}& \text{(3.13)}\end{array}$$ The lattice generated by $b_1,\dots ,b_n$ will be denoted by $B\text{.}$ It is isomorphic to ${\mathbb{Z}}^n$ and acts on ${ℂ}^n$ by translations. Namely, $b\left(u\right)=u+2\pi \sqrt{-1}b,$ where $b\in B\text{.}$ The semi-direct product $\stackrel{\sim }{W}=W⋉B$ is the so-called extended affine Weyl group, acting on ${ℂ}^n$ and leaving $U\prime$ invariant. Picking $u^0\in U\prime ,$ we set $${\pi }_1={\pi }_1(U\prime /\stackrel{\sim }{W},u^0)\text{.}$$ The group structure of ${\pi }_1$ is described as follows. Given an element $w\in \stackrel{\sim }{W},$ let ${\gamma }_w$ be a path from $u^0$ to $w^{-1}\left(u^0\right)$ in $U\prime \text{.}$ For elements $w_1,w_2\in \stackrel{\sim }{W},$ we define the composition ${\gamma }_{w_2}\circ {\gamma }_{w_1}$ of ${\gamma }_{w_1}$ and ${\gamma }_{w_2}$ as the path composed of ${\gamma }_{w_1}$ and the path ${\gamma }_{w_2}$ mapped by $w_1^{-1}$ (see Fig.5). The class of $\gamma$ will be denoted by $\bar{\gamma }\text{.}$ The map ${\bar{\gamma }}_w\to w$ is a homomorphism onto $\stackrel{\sim }{W}\text{.}$

It is convenient to choose $u^0$ and the generators of ${\pi }_1$ as follows. Set $\Re =\text{Re},$ $\Im =\text{Im},$ $$C={\left(\sqrt{-1}ℝ\right)}^n\backslash \{u\in {\left(\sqrt{-1}ℝ\right)}^n\hspace{0.17em}|\hspace{0.17em}0<\Im u_{\alpha }<2\pi \hspace{0.17em}\text{for every}\hspace{0.17em}\alpha \in {\mathrm{\Sigma }}_{+}\}\text{.}$$ Then ${ℂ}^n\backslash C$ is a simply connected open subset of $U\prime \text{.}$ Let us take $u^0\in {ℂ}^n$ such that $\Re u_{\alpha }^0\gg 0$ for $\alpha \in {\mathrm{\Sigma }}_{+}\text{.}$ For any element $w\in \stackrel{\sim }{W},$ we denote a path from $u^0$ to $w^{-1}\left(u^0\right)$ in ${ℂ}^n\backslash C$ by ${\gamma }_w\text{.}$ This condition simply means that whenever $u_{\alpha }\in iℝ$ intersects the imaginary axis it must go through the 'window' $0<\Im u_{\alpha }<2\pi \text{.}$ $$\begin{array}{c}\begin{array}{c} u0 γw1 w1-1(u0) γw2 w1-1 w2-1(u0) w1-1w2-1(u0) \end{array}\\ \text{Figure. 5. Composition of paths}\end{array}$$

For any element $w\in \stackrel{\sim }{W}$ we define an element ${\bar{\gamma }}_w$ of ${\pi }_1$ to be the image of ${\gamma }_w\text{.}$ Since ${ℂ}^n\backslash C$ is simply connected, ${\bar{\gamma }}_w$ depends only on $w\text{.}$ We set ${\tau }_i={\gamma }_{s_i}$ and choose ${\chi }_i$ to be a path from $u^0$ to the point $u\prime$ with the same coordinates $u_j^{\prime }=u_j^0$ for $j\ne i$ and $u_i^{\prime }=u_i^0+2\pi \sqrt{-1}\text{.}$ The structure of ${\pi }_1$ is described in the following theorem from [Lek1981].

Theorem 3.3. $$\begin{array}{cc}{\pi }_1=⟨{\bar{\tau }}_1,\dots ,{\bar{\tau }}_n,{\bar{\chi }}_1,\dots ,{\bar{\chi }}_n⟩,& \text{(3.14)}\\ {\bar{\tau }}_i\hspace{0.17em}\text{satisfy the Coxeter relations,}& \text{(3.15)}\\ [{\bar{\chi }}_i,{\bar{\chi }}_j]=[{\bar{\tau }}_i,{\bar{\chi }}_j]=0(i\ne j)\text{.}& \text{(3.16)}\\ {\bar{\tau }}_i^{-1}{\bar{\chi }}_i{\bar{\tau }}_i^{-1}={\bar{\chi }}_{s_i\left(b_i\right)}\text{.}& \text{(3.17)}\end{array}$$ Here for $b=\sum _{i=1}^n{k}_i{b}_i$ we put $$\begin{array}{cc}{\displaystyle {\bar{\chi }}_b=\prod _{i=1}^n{\bar{\chi }}_i^{k_i}\text{.}}& \text{(3.18)}\end{array}$$

Fig.6 proves the relation (3.17). It shows the $u_i\text{-coordinate}$ only, which is sufficient for this relation.

Let us introduce the affine Hecke algebra ${\mathscr{H}}_{\mathrm{\Sigma }}^t$ associated with a root system $\mathrm{\Sigma }$ as a quotient of the group algebra of ${\pi }_1$ by the quadratic relations.

Definition 3.3. The affine Hecke algebra associated with a root system $\mathrm{\Sigma }$ is an associative $ℂ\text{-algebra}$ generated by $1,$ $T_1,\dots ,T_n,$ $X_1,\dots ,X_n$ with the following relations: $$\begin{array}{cc}{T}_i\hspace{0.17em}\text{satisfy the Coxeter relations.}& \text{(3.19)}\\ [X_i,X_j]=[T_i,X_j]=0i\ne j,& \text{(3.20)}\\ T_i^{-1}{X}_i{T}_i^{-1}=X_{s_i\left(b_i\right)},& \text{(3.21)}\\ (T_i-t)(T_i+t^{-1})=0\text{.}& \text{(3.22)}\end{array}$$ $$\begin{array}{c}\begin{array}{c} 0 2π-1 χsi(bi) χi si(u0) u0 τi τi si(u0)+2π-1bi u0+2π-1bi \end{array}\\ \text{Figure. 6. Proof of the relation (3.17)}\end{array}$$ The monomials $X_b$ are defined as in (3.18), $t\in {ℂ}^{*}\text{.}$ Here and above we mean the homogeneous Coxeter relations: $T_i{T}_j{T}_i\dots =T_j{T}_i{T}_j\dots ,$ $m_{ij}$ factors on each side, where $m_{ij}=2,3,4$ whenever the corresponding vertices in the Dynkin diagram are connected by $0,1,2$ laces.

Let $\mathrm{\Phi }$ be an invertible solution of the AKZ equation associated with $\mathrm{\Sigma },$ defined in a neighborhood of $u^0\text{.}$ Then, for $w\in \stackrel{\sim }{W},$ $w^{-1}\left(\mathrm{\Phi }\right)$ is defined near $w^{-1}\left(u^0\right)$ (see (2.30)). Let $\gamma$ be a path in $U\prime$ from $u^0$ to $w^{-1}\left(u^0\right)\text{.}$ Denote by ${\left(w^{-1}\left(\mathrm{\Phi }\right)\right)}_{\bar{\gamma }}$ the analytic continuation of $w^{-1}\left(\mathrm{\Phi }\right)$ back to $u^0$ along the path $\gamma ,$ where $\bar{\gamma }$ denotes the class of $\gamma$ in the fundamental group ${\pi }_1\text{.}$ We will also use the projection homomorphism $\stackrel{\sim }{W}\to W$ sending $w=\bar{w}b$ to $\bar{w}$ for $b\in B,$ $\bar{w}\in W\text{.}$ Using this homomorphism we can extend the action of $W$ from (2.30) to $\stackrel{\sim }{W},$ multiplying $\mathrm{\Phi }$ on the left by $\bar{w}\text{.}$

Let us define the monodromy $T_{\bar{\gamma }}$ to be the ratio $$\begin{array}{cc}{T}_{\bar{\gamma }}={\left(w^{-1}\left(\mathrm{\Phi }\right)\right)}_{\bar{\gamma }}^{-1}\mathrm{\Phi }={\left(\mathrm{\Phi }\left(w\left(u\right)\right)\right)}_{\bar{\gamma }}^{-1}·\bar{w}·\mathrm{\Phi }\text{.}& \text{(3.23)}\end{array}$$ Here dot means the product in ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ Since $\mathrm{\Phi }$ and $w^{-1}\left(\mathrm{\Phi }\right)$ both satisfy the same AKZ equation, $T_{\bar{\gamma }}$ does not depend on $u\text{.}$ So it is an invariant of the homotopy class of $\gamma$ and is always invertible. If we choose $u^0$ and the paths ${\gamma }_w$ in ${ℂ}^n\backslash C$ as above, then $T_w$ for $w\in \stackrel{\sim }{W}$ are well-defined. The monodromy is a homomorphism from ${\pi }_1$ (but not from $\stackrel{\sim }{W}\text{),}$ which readily results from the definition.

As a preparation for an explicit computation of $\left\{T_w\right\}$ in the next section, we shall introduce a special class of solutions $\mathrm{\Phi }\text{.}$

Proposition 3.4. For generic $\lambda ,$ there exists a unique solution ${\mathrm{\Phi }}_{\text{as}}\left(u\right)$ of the AKZ equation such that $$\begin{array}{cccc}{\mathrm{\Phi }}_{\text{as}}\left(u\right)& =& \hat{\mathrm{\Phi }}\left(u\right)e^{\sum _{i=1}^n{u}_i{x}_i}\text{for}& \text{(3.24)}\\ \hat{\mathrm{\Phi }}\left(u\right)& =& {\displaystyle 1+\sum _{m=(m_1,\dots ,m_n),m_i\ge 0,m\ne 0}{\mathrm{\Phi }}_m{e}^{-\sum _{i=1}^n{m}_i{u}_i},}& \text{(3.25)}\end{array}$$ where $\Re u⟶\infty ,$ and ${\mathrm{\Phi }}_m$ are independent of $u\text{.}$

We call the solution in the proposition the asymptotically free solution. To be more exact, we need either to complete ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime },$ 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 $A_1\text{.}$ A general consideration follows the same lines. In this case, $$\begin{array}{cc}{\displaystyle {\mathrm{\Phi }}_{\text{as}}\left(u\right)=(1+\sum _{m>0}{\mathrm{\Phi }}_m{e}^{-mu})e^{ux}=\hat{\mathrm{\Phi }}\left(u\right)e^{ux}\text{.}}& \text{(3.26)}\end{array}$$ The equation (2.39) leads to $$\begin{array}{cc}{\displaystyle \frac{\partial \hat{\mathrm{\Phi }}\left(u\right)}{\partial u}=k\frac{s}{e^u-1}\hat{\mathrm{\Phi }}\left(u\right)+[x,\hat{\mathrm{\Phi }}\left(u\right)]\text{.}}& \text{(3.27)}\end{array}$$ Comparing the coefficients of $e^{-mu}\text{:}$ $$\begin{array}{cc}{\displaystyle -m{\mathrm{\Phi }}_m=[x,{\mathrm{\Phi }}_m]+\text{(}\text{terms with}\hspace{0.17em}{\mathrm{\Phi }}_j,j>m\text{).}}& \text{(3.28)}\end{array}$$ Given a representation of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime },$ we find ${\mathrm{\Phi }}_m$ assuming that $m+\text{ad}\left(x\right)$ is invertible for any $m>0$ in this representation. Therefore, setting $\text{Spec}\left(x\right)=\left\{{\mu }_j\right\},$ the conditions $m+{\mu }_i-{\mu }_j\ne 0,$ $m=1,2,\dots ,$ 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 ${\mathscr{H}}_{\mathrm{\Sigma }}^t$ and ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ using the monodromy of the AKZ equation.

Let us fix an invertible solution $\mathrm{\Phi }\left(u\right)$ of the AKZ system in a neighborhood of $u^0\in U^{*}={ℂ}^n\backslash C\subset U\prime \text{.}$ The functions $\mathrm{\Phi }\left(w\left(u\right)\right)$ will be extended to $u^0$ through $U^{*}\text{.}$ Since ${\mathscr{H}}_{\mathrm{\Sigma }}$ 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 ${\mathscr{H}}_{\mathrm{\Sigma }}^t$ to ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ given by $$T_j⟼T_j^{\prime },X_j⟼X_j^{\prime },$$ where $$T_j^{\prime }=\mathrm{\Phi }{\left(s_j\left(u\right)\right)}^{-1}{s}_j\mathrm{\Phi }\left(u\right),X_j^{\prime }=\mathrm{\Phi }{(u-2\pi \sqrt{-1}{b}_j)}^{-1}\mathrm{\Phi }\left(u\right)\text{.}$$ If $t=\text{exp}\left(\pi \sqrt{-1}k\right)$ 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), $T_j^{\prime }=T_{{\bar{\tau }}_j}$ and $X_j^{\prime }=T_{{\bar{\chi }}_j}\text{.}$ 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 $T_j^{\prime }$ and $X_j^{\prime }$ for the asymptotically free solution ${\mathrm{\Phi }}_{\text{as}}\left(u\right)\text{.}$ Given $b\in B,$ we set $$X_b=\prod _{j=1}^n{X}_j^{k_j}\text{for}\hspace{0.17em}b=\sum _{j=1}^n{k}_j{b}_j,$$ and define $X_b^{\prime }$ analogously.

Theorem 3.6 ([Che1991-3]). Let us choose the asymptotically free solution ${\mathrm{\Phi }}_{\text{as}}\left(u\right)$ as $\mathrm{\Phi }\left(u\right)\text{.}$ Then

(a)	$X_j^{\prime }=\text{exp}\left(2\pi \sqrt{-1}{x}_j\right),$
(b)	$s_i-\frac{k}{x_{{\alpha }_i}}=g\left(x_{{\alpha }_i}\right)(T_i^{\prime }+\frac{t-t^{-1}}{{X_a^{\prime }}^{-1}-1}),$

where the function $g\left(v\right)$ is defined by $$g\left(v\right)=\frac{\mathrm{\Gamma }{(1+v)}^2}{\mathrm{\Gamma }(1+k+v)\mathrm{\Gamma }(1-k+v)},$$ and (b) is in fact a formula for $T_i^{\prime }$ in terms of $\{s,x\}\text{.}$

We will give a sketch of the proof of Theorem 3.6. The statement (a) is immediate, since $${\mathrm{\Phi }}_{\text{as}}(u-2\pi \sqrt{-1}{b}_j)=\hat{\mathrm{\Phi }}\left(u\right)e^{\sum u_i{x}_i-2\pi \sqrt{-1}{x}_j}={\mathrm{\Phi }}_{\text{as}}\left(u\right)e^{-2\pi \sqrt{-1}{x}_j}\text{.}$$ To prove the statement (b), we reduce the problem to the $A_1$ case. Let us fix the index $i$ $\text{(}1\le i\le n\text{).}$ Set $E\left(u\right)=e^{\sum _{i=1}^n{u}_i{x}_i},$ so that ${\mathrm{\Phi }}_{\text{as}}\left(u\right)=\hat{\mathrm{\Phi }}\left(u\right)E\left(u\right)\text{.}$ Let us define ${\mathrm{\Phi }}^{\left(i\right)}\left(u\right)$ as follows: $${\mathrm{\Phi }}^{\left(i\right)}\left(u\right)={\mathrm{\Phi }}^{\infty \left(i\right)}\left(u_i\right)E\left(u\right),$$ where ${\mathrm{\Phi }}^{\infty \left(i\right)}\left(u_i\right)=\underset{\Re u_j\to +\infty (j\ne i)}{\text{lim}}\hat{\mathrm{\Phi }}\left(u\right)\text{.}$ The AKZ system for ${\mathrm{\Phi }}^{\left(i\right)}\left(u\right)$ reads: $$\begin{array}{cccc}{\displaystyle \frac{\partial {\mathrm{\Phi }}^{\left(i\right)}}{\partial u_i}}& =& {\displaystyle (k\frac{s_i}{e^{u_i}-1}+x_i){\mathrm{\Phi }}^{\left(i\right)},}& \text{(3.29)}\\ {\displaystyle \frac{\partial {\mathrm{\Phi }}^{\left(i\right)}}{\partial u_j}}& =& {\displaystyle x_j{\mathrm{\Phi }}^{\left(i\right)}(j\ne i)\text{.}}& \text{(3.30)}\end{array}$$ Reduction procedure. Since the monodromy $T_i^{\prime }$ does not depend on $u,$ the point $u^0$ and the path connecting $u^0$ and $s_i\left(u^0\right)$ may be replaced by any deformations in $U\prime$ or their limits. Provided the existence, the resulting monodromy coincides with $T_i^{\prime }\text{.}$ For instance, $T_i^{\prime }$ equals $$T_i^{\left(i\right)}={\left({\mathrm{\Phi }}^{\left(i\right)}\left(s_i\left(u\right)\right)\right)}^{-1}{s}_i{\mathrm{\Phi }}^{\left(i\right)}\left(u\right)\text{.}$$

Indeed, the latter is the limiting monodromy for a path with $\Re u_j$ $\text{(}j\ne i\text{)}$ approaching the infinity. We note that $\Re u_j\left(s_i\left(u^0\right)\right)\to +\infty$ if $\Re u_j\left(u^0\right)$ does.

In the reduced equations (3.29) and (3.30), we may diminish the values, considering the subalgebra of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ generated by $x_j$ $\text{(}1\le j\le n\text{),}$ and $s_i\text{.}$ In this algebra, the following elements are central: $$x_j(j\ne i),x_i-\frac{1}{2}{x}_{{\alpha }_i}\text{.}$$ Hence, if we define $E^{\left(i\right)}\left(u\right)$ by $$E^{\left(i\right)}\left(u\right)=e^{\sum _{j=1}^n{u}_j{x}_j-u_i{x}_{{\alpha }_i}/2},$$ it enjoys the following properties:

(i)	$E^{\left(i\right)}\left(u\right)$ commutes with $s_i,$
(ii)	$E^{\left(i\right)}\left(s_i\left(u\right)\right)=E^{\left(i\right)}\left(u\right)\text{.}$

The second property can be verified directly: $$\sum _j{\left(s_i\left(u\right)\right)}_j{x}_j-\frac{1}{2}{\left(s_i\left(u\right)\right)}_i{x}_{{\alpha }_i}=\sum _j(u_j-(a_i,{\alpha }_j)u_i)x_j+\frac{1}{2}{u}_i{x}_{a_i}=\sum _j{u}_j{x}_j-\frac{1}{2}{u}_i{x}_{a_i}\text{.}$$ We have used that ${\left(s_i\left(u\right)\right)}_j=({\alpha }_j,s_i\left(u\right))$ and $\sum _j(a_i,{\alpha }_j)x_j=x_{a_i}\text{.}$ Setting $${\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right)={\mathrm{\Phi }}^{\left(i\right)}\left(u\right)E^{\left(i\right)}{\left(u\right)}^{-1},$$ the system of equations (3.29),(3.30) becomes precisely the AKZ equation for ${\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right)$ in the $A_1$ case: $$\begin{array}{cccc}{\displaystyle \frac{\partial {\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right)}{\partial u_i}}& =& {\displaystyle (k\frac{s_i}{e^{u_i}-1}+\frac{1}{2}{x}_{a_i}){\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right),}& \text{(3.31)}\\ {\displaystyle \frac{\partial {\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right)}{\partial u_j}}& =& {\displaystyle 0(j\ne i)\text{.}}& \text{(3.32)}\end{array}$$ Because of the above properties of $E^{\left(i\right)}\left(u\right),$ the monodromy of ${\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right)$ coincides with $T_i^{\prime }\text{.}$ However ${\stackrel{\sim }{\mathrm{\Phi }}}^{\left(i\right)}\left(u\right)$ 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 $T\prime ,$ let us involve the intertwiners of ${\mathscr{H}}_{\mathrm{\Sigma }}^t\text{.}$ They are defined similar to those in the degenerate case: $$f_i=s_i-\frac{k}{x_{a_i}}\hspace{0.17em}\text{for}\hspace{0.17em}{\mathscr{H}}_{\mathrm{\Sigma }}^{\prime },F_i=T_i+\frac{t-t^{-1}}{X_{a_i}^{-1}-1}\hspace{0.17em}\text{for}\hspace{0.17em}{\mathscr{H}}_{\mathrm{\Sigma }}^t\text{.}$$

Lemma 3.7. $F_i{X}_b=X_{s_i\left(b\right)}{F}_i\text{.}$

It readily results from the definition of ${\mathscr{H}}_{\mathrm{\Sigma }}^t$ (cf. 3.10).

The image $F_i^{\prime }$ of $F_i$ in ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ with respect to the homomorphism constructed in Theorem 3.5 can be represented as $F_i^{\prime }=g_i\left(x\right)f_i$ for a function $g_i$ of $x\text{.}$ Indeed, $f_i{X}_b^{\prime }=X_{s_i\left(b\right)}^{\prime }{f}_i,$ which gives the proportionality. Recall that $X_b^{\prime }=\text{exp}\left(2\pi \sqrt{-1}{x}_b\right)\text{.}$ Here $g_i\left(x\right)$ must be of the form $g\left(x_{a_i}\right)$ 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 $T_i^{\prime }$ 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 $z_1,z_2$ and a parameter $z_0\text{:}$ $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }\prime }{\partial z_j}=[k\left(\frac{s_1}{z_j-z_k}\right)+\frac{{\mathrm{\Omega }}_j}{z_j-z_0}]\mathrm{\Phi }\prime (j=1,2,k=3-j)\text{.}}& \text{(3.33)}\end{array}$$ When $z_0=0$ the substitutions are as follows $$2x_1={\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2+k{s}_1,u_1=\text{log}(z_1/z_2),\mathrm{\Phi }\prime ={\mathrm{\Phi }}^{\left(1\right)}\left(u_1\right){\left(z_1{z}_2\right)}^{-1/2({\mathrm{\Omega }}_1+{\mathrm{\Omega }}_2+k{s}_1)}\text{.}$$ The monodromy corresponding to the transposition of $z_1$ and $z_2$ for $z_0=0$ coincides with $T_1^{\prime }\text{.}$ It does not depend on $z_0$ up to a conjugation (the same reduction argument applied to the KZ-equation with three variables). Sending $z_0$ to infinity we eliminate the $\mathrm{\Omega }\text{-terms.}$ The monodromy of the resulting equation can be calculated immediately. Since it is conjugated to $T_1^{\prime }$ 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 ${\mathscr{H}}^t$ is the same, we readily conclude that these equations are isomorphic for generic $\lambda \text{.}$ This will be made much more constructive below. We will also consider any $\lambda \text{.}$

Remark 3.2. Let us apply Theorem 3.6 to the standard rational KZ equation in the ${GL}_n$ case. We calculated the monodromy of $$\frac{\partial \mathrm{\Phi }}{\partial v_i}=(k\sum _{j>i}\frac{s_{ij}}{e^{v_i-v_j}-1}-k\sum _{j<i}\frac{s_{ij}}{e^{v_j-v_i}-1}+y_i)\mathrm{\Phi }(1\le i\le n)\text{.}$$ Taking special $y_i=k\sum _{j=i+1}^n{s}_{ij}$ and substituting $z_i=e^{v_i},$ we come to $$\frac{\partial \mathrm{\Phi }}{\partial z_i}=k\sum _{j\ne i}\frac{s_{ij}}{z_i-z_j}\mathrm{\Phi }(1\le i\le n)\text{.}$$ It corresponds to the simplest ${\mathrm{\Omega }}_{ij}=0$ in (2.18). By the way, these $\left\{y\right\}$ induce a homomorphism from ${\mathscr{H}}_n^{\prime }$ to $ℂ{𝕊}_{n+1}$ due to Drinfeld. Diagonalizing the commuting elements $\sum _{j>i}{s}_{ij}$ 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 $${\mathscr{H}}_{\mathrm{\Sigma }}^t\simeq {\mathscr{H}}_{\mathrm{\Sigma }}^{\prime },X_j⟼t^{2x_j},$$ where $t=e^{\pi \sqrt{-1}k}$ and represented it as a relation between the intertwiners of the degenerate and non-degenerate affine Hecke algebras: $$F_j=T_j+\frac{t-t^{-1}}{X_{a_j}^{-1}-1}⟼g\left(x_{a_j}\right)(s_j-\frac{k}{x_{a_j}})\text{.}$$ This construction can be naturally generalized. In fact we need only a very mild restriction on $g\left(x\right)$ to get such a homomorphism. Normalizing the intertwiners to make them 'unitary' $\text{(}{f}^2=1=F^2\text{),}$ we come to the simplest possible map: $$X_j⟼t^{2x_j},\frac{F_j}{t+\frac{t-t^{-1}}{X_{a_j}^{-1}-1}}⟼\frac{s_j-\frac{k}{x_{a_j}}}{1-\frac{k}{x_{a_j}}}\text{.}$$

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: $${}^w\mathrm{\Phi }\left(u\right)=\mathrm{\Phi }\left(w^{-1}\left(u\right)\right),w=\bar{w}b\in \stackrel{\sim }{W}=W⋉B,u\in {ℂ}^n\text{.}$$

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

(i)	$U$ does not contain 'bad hyperplanes' $\prod _{\alpha \in {\mathrm{\Sigma }}_{+}}(e^{u_{\alpha }}-1)=0,$
(ii)	$U$ is simply connected,
(iii)	$({ℂ}^n\backslash \underset{w\in \stackrel{\sim }{W}}{\cup }w\left(C\right))/\stackrel{\sim }{W}$ is connected.

We shall refer to such $C$ as a system of cutoffs. In §3.2, a special system of cutoffs $\left(U^{*}\right)$ has been already used in order to compute the monodromy.

Let us fix a system of cutoffs $C$ and $U\text{.}$ Then for each $w\in \stackrel{\sim }{W}$ there is a path ${\gamma }_w$ (unique up to homotopy) joining $u^0$ and $w^{-1}\left(u^0\right)\text{.}$ So the choice of $C$ implies a choice of representatives ${\bar{\gamma }}_w$ in the fundamental group ${\pi }_1(U\prime /\stackrel{\sim }{W},u^0)\text{.}$ Here $U\prime$ is the complement of the union of 'bad hyperplanes' (3.13).

We pick a solution $\mathrm{\Phi }$ of the AKZ equation in $U$ and define the monodromy function ${𝒯}_w$ $\text{(}w\in \stackrel{\sim }{W}\text{)}$ $$\begin{array}{cc}\bar{w}\mathrm{\Phi }={}^{w^{-1}}\mathrm{\Phi }·{𝒯}_{w}w=\bar{w}b\in \stackrel{\sim }{W}\text{.}& \text{(3.34)}\end{array}$$ Here $\mathrm{\Phi }$ is invertible at least at one point and is extended analytically to the whole $U\text{.}$ The values are in the endomorphisms of any finite-dimensional representation of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ (we will apply the construction to the induced representations).

The monodromy ${\left\{{𝒯}_w\right\}}_{w\in \stackrel{\sim }{W}}$ satisfies the following:

(a)	(1-cocycle condition) ${}^{v^{-1}}\left({𝒯}_w\right){𝒯}_v={𝒯}_{wv}\forall w,v\in \stackrel{\sim }{W},$
(b)	$\frac{\partial }{\partial u_i}{𝒯}_w=0,$ and hence ${𝒯}_w$ is locally constant.

The property (b) holds since both $\mathrm{\Phi }$ and $w\left(\mathrm{\Phi }\right)={\bar{w}}^w\mathrm{\Phi }$ satisfy the same differential equation of the first order (the AKZ equation). It readily results in the invertibility of ${𝒯}_w$ on ${ℂ}^n-\underset{w\in \stackrel{\sim }{W}}{\cup }w\left(C\right)\text{.}$ The latter set is not connected, so $𝒯$ is not just a constant.

Next let us introduce the operators ${\sigma }_w,{\sigma }_w^{\prime }$ $\text{(}w\in \stackrel{\sim }{W}\text{),}$ acting on functions $F$ on $U\text{:}$ $$\begin{array}{ccc}\left({\sigma }_{w}F\right)\left(u\right)& =& \left({}^{w^{-1}}F\right)\left(u\right)=F\left(w\left(u\right)\right),{\sigma }_i={\sigma }_{s_i},\\ \left({\sigma }_w^{\prime }F\right)\left(u\right)& =& \left({}^{w^{-1}}F\right)\left(u\right){𝒯}_w,{\sigma }_i^{\prime }={\sigma }_{s_i}^{\prime }\text{.}\end{array}$$ The relations for the operators ${\sigma }_w^{\prime }$ are the same as for the permutations ${\sigma }_w\text{:}$ $$\begin{array}{cc}\begin{array}{c}\text{a)}{\sigma }_w^{\prime }{\sigma }_v^{\prime }={\sigma }_{vw}^{\prime },\\ \text{b)}{\sigma }_w^{\prime }{u}_b=u_{w^{-1}\left(b\right)}{\sigma }_w^{\prime },\\ \text{c)}{\sigma }_w^{\prime }{\partial }_b={\partial }_{w^{-1}\left(b\right)}{\sigma }_w^{\prime },{\partial }_b\left(u_{\alpha }\right)=(b,\alpha )\text{.}\end{array}& \text{(3.35)}\end{array}$$ Note that the property a) follows from the $1\text{-cocycle}$ condition for ${\left\{{𝒯}_w\right\}}_{w\in \stackrel{\sim }{W}}\text{.}$ Indeed, $$\begin{array}{ccc}\left({\sigma }_w^{\prime }{\sigma }_v^{\prime }\right)\left(F\right)& =& {\sigma }_w^{\prime }\left({\sigma }_v^{\prime }\left(F\right)\right)\\ & =& {\sigma }_w^{\prime }\left({}^{v^{-1}}F{𝒯}_v\right)\\ & =& {}^{w^{-1}}\left({}^{v^{-1}}F{𝒯}_v\right){𝒯}_w\\ & =& {}^{w^{-1}}{}^{v^{-1}}F\left({}^{w^{-1}}{𝒯}_v\right){𝒯}_w\\ & =& {}^{{\left(vw\right)}^{-1}}F{𝒯}_{vw}\\ & =& {\sigma }_{vw}^{\prime }\left(F\right)\text{.}\end{array}$$

Let ${𝒮\text{ol}}_{\text{AKZ}}$ be the space of solutions of the AKZ equation with values in ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ When we consider the AKZ equation on a finite-dimensional ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-module}$ $V,$ we will denote the space of its solutions by ${𝒮\text{ol}}_{\text{AKZ}}\left(V\right)\text{.}$ Starting with AKZ let us go to QMBP. In what follows, $\mathrm{\Phi }\in {𝒮\text{ol}}_{\text{AKZ}}$ or $\mathrm{\Phi }\in {𝒮\text{ol}}_{\text{AKZ}}\left(\text{End}\left(V\right)\right)\text{.}$ In the latter case all operators act on $\text{End}\left(V\right)\text{-valued}$ functions.

(1)	Using $s_{\alpha }\mathrm{\Phi }={\sigma }_{s_{\alpha }}^{\prime }\mathrm{\Phi },$ we rewrite the AKZ equation: $$\begin{array}{ccc}{x}_i\mathrm{\Phi }& =& {\displaystyle (\frac{\partial }{\partial u_i}-k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^i\frac{s_{\alpha }}{e^{u_{\alpha }}-1})\mathrm{\Phi }}\\ & =& {\displaystyle (\frac{\partial }{\partial u_i}-k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^i{(e^{u_{\alpha }}-1)}^{-1}{\sigma }_{s_{\alpha }}^{\prime })\mathrm{\Phi }(1\le i\le n)\text{.}}\end{array}$$ Let us denote: $${𝒟}_i^{\prime }=\frac{\partial }{\partial u_i}-k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^i{(e^{u_{\alpha }}-1)}^{-1}{\sigma }_{s_{\alpha }}^{\prime }\text{.}$$ The local invertibility of $\mathrm{\Phi }$ and the relations ${𝒟}_i^{\prime }\mathrm{\Phi }=x_i\mathrm{\Phi }$ result in the commutativity $$[{𝒟}_i^{\prime },{𝒟}_j^{\prime }]=0\forall i,j\text{.}$$ Here one can use that the commutators do not contain the derivatives, which readily results from the relations for $\sigma \prime \text{.}$ 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 $x_i$ commutes with ${𝒟}_j^{\prime },$ we get $$p(x_1,\dots ,x_n)\mathrm{\Phi }=p({𝒟}_1^{\prime },\dots ,{𝒟}_n^{\prime })\mathrm{\Phi }$$ for any polynomial $p\in ℂ[x_1,\dots ,x_n]\text{.}$
(3)	For $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {ℂ}^n$ let us take an ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-module}$ $V_{\lambda }$ with the following properties: $$\begin{array}{}\begin{array}{cc}\text{(i)}p(x_1,\dots ,x_n)=p({\lambda }_1,\dots ,{\lambda }_n)\text{on}\hspace{0.17em}{V}_{\lambda }\hspace{0.17em}\text{for any}\hspace{0.17em}p\in ℂ{[x_1,\dots ,x_n]}^W,& \text{(3.36)}\\ \text{(ii)}\text{there exists a linear map}\hspace{0.17em}\text{tr}:V_{\lambda }⟶ℂ\hspace{0.17em}\text{satisfying}& \\ \text{tr}\left(wa\right)=\text{tr}\left(a\right)\forall w\in W,a\in V_{\lambda }\text{.}& \text{(3.37)}\end{array}\end{array}$$

Let $p(x_1,\dots ,x_n)$ be a polynomial. Using the commutation relations (3.35), we can write $$p({𝒟}_1^{\prime },\dots ,{𝒟}_n^{\prime })=\sum _{w\in W}{D\prime }_w^{\left(p\right)}{\sigma }_w^{\prime },$$ where ${𝒟\prime }_w^{\left(p\right)}$ are differential operators (they do not contain $\sigma \prime \text{).}$ They are scalar and commute with ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$ Thus $$p(x_1,\dots ,x_n)\mathrm{\Phi }=\sum _{w\in W}{𝒟\prime }_w^{\left(p\right)}{\sigma }_w^{\prime }\mathrm{\Phi }=\sum _{w\in W}{𝒟\prime }_w^{\left(p\right)}w\mathrm{\Phi }\text{.}$$ Now, we assume that $p$ is $W\text{-invariant.}$ Applying tr (see (3.36) and (3.37)), we come to $$p({\lambda }_1,\dots ,{\lambda }_n)\psi =L_p^{\prime }\psi \hspace{0.17em}\text{for}\hspace{0.17em}{L}_p^{\prime }=\sum _{w\in W}{𝒟\prime }_w^{\left(p\right)},$$ where $$\psi \left(u\right)=\text{tr}\left(\mathrm{\Phi }\left(u\right)\right)$$ is a $ℂ\text{-valued}$ function. The differential operators $L_p^{\prime }$ are $W\text{-invariant,}$ which follows from the same construction (we will reprove this algebraically below).

Let us introduce the trigonometric Dunkl operators ${𝒟}_i$ $\text{(}1\le i\le n\text{)}$ replacing $\sigma \prime$ by $\sigma \text{:}$ $${𝒟}_i=\frac{\partial }{\partial u_i}-k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^i{(e^{u_{\alpha }}-1)}^{-1}{\sigma }_{s_{\alpha }}\text{.}$$ Repeating the above construction, define ${𝒟}^{\left(p\right)}$ for a $W\text{-invariant}$ polynomial $p$ by $$p({𝒟}_1,\dots ,{𝒟}_n)=\sum _{w\in W}{𝒟}_w^{\left(p\right)}{\sigma }_w\text{.}$$ Since in the construction of $L_p^{\prime }$ and $L_p$ we use only the commutation relations (3.35) for ${\sigma }_w^{\prime }$ and ${\sigma }_i$ 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 $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {ℂ}^n\text{.}$ It is the following system of differential equations for a $ℂ\text{-valued}$ function $\psi \text{.}$ $$L_p\psi =p({\lambda }_1,\dots ,{\lambda }_n)\psi (p\in ℂ{[x_1,\dots ,x_n]}^W)\text{.}$$ It is known [HOp1987] (and easy to see by looking at the leading terms of $L_p\text{)}$ that the dimension of the space of solutions $\psi$ is $\left|W\right|\text{.}$

Summarizing, we come to the theorem.

Theorem 3.8. Applying $\text{tr}$ we get a homomorphism $$\text{tr}:{𝒮\text{ol}}_{\text{AKZ}}\left(V_{\lambda }\right)⟶{𝒮\text{ol}}_{\text{QMBP}}\left(\lambda \right)\text{.}$$ Here ${𝒮\text{ol}}_{\text{QMBP}}\left(\lambda \right)$ denotes the space of solutions to QMBP with the eigenvalue $\lambda \text{.}$

We can say more for concrete represenatations, especially for the induced representations $J_{\lambda }^{\circ }$ (see (3.4)). We define the 'trace' $$\text{tr}:J_{\lambda }^{\circ }⟶ℂ$$ as the map dual to the embedding $$ℂ⟶J_{\lambda }=ℂ[x_1,\dots ,x_n]/L_{\lambda }$$ sending $1$ to $1\in ℂ[x_1,\dots ,x_n]\text{.}$ Here $L_{\lambda }$ denotes the ideal generated by $p\left(x\right)-p\left(\lambda \right),$ $p\in ℂ{[x_1,\dots ,x_n]}^W\text{.}$ One easily checks that tr satisfies the conditions (3.37).

Theorem 3.9 ([Che1991-4]) For any $\lambda \in {ℂ}^n,$ $\text{tr}$ gives an isomorphism $$\text{tr}:{𝒮\text{ol}}_{\text{AKZ}}\left(J_{\lambda }^{\circ }\right)\stackrel{\sim }{⟶}{𝒮\text{ol}}_{\text{QMBP}}\left(\lambda \right)\text{.}$$

		Proof.
	The key observation: $$\begin{array}{cc}\text{for any}\hspace{0.17em}{\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{-submodule}\hspace{0.17em}0\ne M\subset J_{\lambda }^{\circ }\text{, we have}\hspace{0.17em}\text{tr}{|}_M\ne 0\text{.}& \text{(3.38)}\end{array}$$ Indeed, if $0\ne f\in M,$ then there exists a polynomial $p\left(x\right)\in ℂ[x_1,\dots ,x_n]$ such that $f\left(p\right)\ne 0\text{.}$ However $f\left(p\right)=\text{tr}\left(p\left(f\right)\right)\in \text{tr}\left(M\right)\text{.}$ 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 $\left|W\right|\text{).}$ So let us suppose that for $\phi \left(u\right)\in {𝒮\text{ol}}_{\text{AKZ}}\left(J_{\lambda }^{\circ }\right)$ identically $$\begin{array}{cc}\text{tr}\left(\phi \right)=0\text{.}& \text{(3.39)}\end{array}$$ We will show that $$\begin{array}{cc}\text{tr}\left({\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\phi \right)=0\text{.}& \text{(3.40)}\end{array}$$ Differentiating (3.39), $$0=\text{tr}\left(\frac{\partial \phi }{\partial u_i}\right)=k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^i\text{tr}\left(\frac{s_{\alpha }}{e^{u_{\alpha }}-1}\phi \right)+\text{tr}\left(x_i\phi \right)\text{.}$$ By the $W\text{-invariance}$ of $\text{tr},$ $\text{tr}\left(s_{\alpha }\phi \right)=\text{tr}\left(\phi \right)=0\text{.}$ Hence $$\begin{array}{cc}\text{tr}\left(x_i\phi \right)=0\text{.}& \text{(3.41)}\end{array}$$ Differentiating this equation by $u_j$ we have $$0=k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^j\text{tr}\left(x_i\frac{s_{\alpha }}{e^{u_{\alpha }}-1}\phi \right)+\text{tr}\left(x_i{x}_j\phi \right)\text{.}$$ Using the commutation relations of $x_j$ and $s_{\alpha },$ we deduce from (3.39), (3.41) that $$\text{tr}\left(x_i{x}_j\phi \right)=0\text{.}$$ Proceeding in the same way, we establish that $$\text{tr}\left(x_{i_1}\cdots x_{i_l}\phi \right)=0$$ for any $i_1,\dots ,i_l\text{.}$ Combining this with the $W\text{-invariance}$ of tr, we get (3.40). For each $u^0,$ consider the submodule $M={\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\phi \left(u^0\right)\subset J_{\lambda }^{\circ }\text{.}$ Then $\text{tr}{|}_M=0,$ and from the key observation above, we deduce that $M=0\text{.}$ This completes the proof of Theorem 3.9. $\square$

The map from Theorem 3.9 was found by Matsuo [Mat1992] for induced representations $I_{\lambda }\text{.}$ He proved his theorem algebraically (without the passage through the trigonometric Dunkl operators discussed above) using an explicit presentation for AKZ in $I_{\lambda }\text{.}$ The isomorphism for $J_{\lambda }^{\circ }$ (or for $I_{\lambda }$ with properly ordered $\lambda$ - (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 $\lambda \text{.}$ I used the modules $J\text{.}$ 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 $L_p\text{.}$

Example 3.1. Let $p_2(x_1,\dots ,x_n)=\sum _{i=1}^n{x}_{{\alpha }_i}{x}_i\text{.}$ Then we have $$L_2=L_{p_2}=\sum _{i=1}^n{\partial }_{{\alpha }_i}{\partial }_i+\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}(\alpha ,\alpha )\frac{k(1-k)}{{(e^{u_{\alpha }}-e^{-u_{\alpha }})}^2}+\text{const.}$$ It was studied in [OPe1983].

Remark 3.4. More generally, let $A$ be a $ℂ\left[W\right]\text{-module}$ and $$V_{A,\lambda }={({\text{Ind}}_{ℂ\left[W\right]}^{{\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }}\left(A\right)/L_{\lambda })}^{\circ }\text{.}$$ As before, $L_{\lambda }$ is the ideal generated by $p\left(x\right)-p\left(\lambda \right),$ $p\in ℂ{[x_1,\dots ,x_n]}^W\text{.}$ Then the following holds $${𝒮\text{ol}}_{\text{AKZ}}\left(V_{A,\lambda }\right)\stackrel{\sim }{⟶}{{𝒮\text{ol}}_{\text{QMBP}}}_A\left(\lambda \right)$$ 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 $L\text{-operator}$ corresponding to $p_2$ above reads $$L_2=\sum _{i=1}^n{\partial }_{{\alpha }_i}{\partial }_i+\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}(\alpha ,\alpha )\frac{k(s_{\alpha }^{*}-k)}{{(e^{u_{\alpha }}-e^{-u_{\alpha }})}^2}+\text{const,}$$ where by $s_{\alpha }^{*}$ we mean the image of $s_{\alpha }$ in $\text{Aut}\left(A\right)\text{.}$

The ${GL}_n$ case

Let us describe AKZ and QMBP in the ${GL}_n$ case.

In §2.4, we introduced the degenerate affine Hecke algebra of type ${GL}_n\text{.}$ It is the algebra $${\mathscr{H}}_n^{\prime }=⟨ℂ{𝕊}_n,y_1,\dots ,y_n⟩$$ subject to the following relations: $$\begin{array}{c}{s}_i{y}_i-y_{i+1}{s}_i=k,s_i{y}_j=y_j{s}_i(i\ne j,j+1),\\ y_i{y}_j=y_j{y}_i(1\le i,j\le n)\text{.}\end{array}$$ As in §2.4, we will use the coordinates $v_i\text{.}$

To prepare the passage to the difference case, we conjugate the AKZ for ${GL}_n$ by the function ${\mathrm{\Delta }}^k$ for A $\mathrm{\Delta }=\prod _{i<j}(e^{v_i}-e^{v_j})\text{.}$ The equation becomes as follows: $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial v_i}=(k(\sum _{j(>i)}\frac{s_{ij}-1}{e^{v_i-v_j}-1}-\sum _{j(<i)}\frac{s_{ij}-1}{e^{v_j-v_i}-1})+y_i+k(i-\frac{n+1}{2}))\mathrm{\Phi }\text{.}}& \text{(3.42)}\end{array}$$ Only in this form it can be quantized (see §4.2). The system is consistent and ${𝕊}_n\text{-invariant.}$

The corresponding Dunkl operators are given by the formula $${𝒟}_i=\frac{\partial }{\partial v_i}-k(\sum _{j(>i)}{(e^{v_i-v_j}-1)}^{-1}({\sigma }_{ij}-1)-\sum _{j(<i)}{(e^{v_j-v_i}-1)}^{-1}({\sigma }_{ij}-1)+i-\frac{n+1}{2})\text{.}$$ Here ${\sigma }_{ij}$ stands for the transpositions of the coordinates: $${\sigma }_{ij}{v}_i=v_j{\sigma }_{ij}\text{.}$$ Similarly, ${\sigma }_w$ means the permutation of the coordinates corresponding to $w^{-1}\text{.}$

The main point of the theory is that they satisfy the relations from the degenerate Hecke algebra: $$[{𝒟}_i,{𝒟}_j]=0=[{𝒟}_i,y_j],\hspace{0.17em}i\ne j,{\sigma }_{i\hspace{0.17em}i+1}{𝒟}_i-{𝒟}_{i+1}{\sigma }_{i\hspace{0.17em}i+1}=k\text{.}$$ 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 $p({𝒟}_1,\dots ,{𝒟}_n)$ and the corresponding $L_p$ are $W\text{-invariant}$ for the $W\text{-invariant}$ polynomials. Use the description of the center of $\mathscr{H}\prime$ to see this.

In the case of ${GL}_n,$ given symmetric $p\in ℂ{[x_1,\dots ,x_n]}^{{𝕊}_n},$ $$p({𝒟}_1,\dots ,{𝒟}_n)=\sum _{w\in {𝕊}_n}{D}_w^{\left(p\right)}{\sigma }_w,$$ where $D_w^{\left(p\right)}$ are scalar differential operators, $$L_p=p({𝒟}_1,\dots ,{𝒟}_n){|}_{\text{symm.poly.}}=\sum _{w\in {𝕊}_n}{D}_w^{\left(p\right)}\text{.}$$ Let us take the elementary symmetric polynomials: $$e_w\left(x\right)=\sum _{i_1<\cdots <i_m}{x}_{i_1}\cdots x_{i_m},$$ as $p,$ setting $L_m=L_{e_m}\text{.}$ Clearly $$L_1=\sum _{i=1}^n\frac{\partial }{\partial v_i}\text{.}$$ The next operator is: $$L_2=\sum _{i<j}\frac{{\partial }^2}{\partial v_i\partial v_j}-\frac{k}{2}\sum _{i<j}\text{coth}\left(\frac{v_i-v_j}{2}\right)(\frac{\partial }{\partial v_i}-\frac{\partial }{\partial v_j})+\frac{k^2}{4}\text{.}$$

When we replace $e_2$ by $p=\sum _i{x}_i^2,$ the corresponding $L\text{-operator}$ is conjugated (by ${\mathrm{\Delta }}^k\text{)}$ 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 $W\text{-valued}$ KZ from [Che1989] by the $x\text{-s).}$ Here the $J_{\lambda }\text{-modules}$ cannot be represented as $I_{\lambda },$ the theorem holds in terms of $J$ only (see [Che1994-3]).

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