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.

The affine Knizhnik-Zamolodchikov equation

We introduce the degenerate affine Hecke algebra and the corresponding affine Knizhnik-Zamolodchikov equation. We show how the former appear as the consistency and invariance conditions for the latter.

The algebra ${\mathscr{H}}_{A_1}^{\prime }$ and the hypergeometric equation

In this section, we introduce the affine Knizhnik-Zamolodchikov (AKZ) equation associated with the root system of type $A_1\text{.}$ It is a first-order differential equation for $\mathrm{\Phi },$ where $\mathrm{\Phi }$ depends on a single variable $u$ and takes values in an infinite-dimensional algebra called the degenerate affine Hecke algebra.

The equation is as follows: $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u}=(k\frac{s}{e^u-1}+x)\mathrm{\Phi }\text{.}}& \text{(2.1)}\end{array}$$ Here $k\in ℂ$ is a parameter, and $s$ and $x$ are operators acting on a vector space where $\mathrm{\Phi }$ takes its value. We impose the following two relations. $$\begin{array}{cc}{s}^2=1,& \text{(2.2)}\\ sx+xs=k\text{.}& \text{(2.3)}\end{array}$$ These relations make (2.1) invariant. Namely, if $\mathrm{\Phi }$ solves (2.1), then $$\begin{array}{cc}\stackrel{\sim }{\mathrm{\Phi }}\left(u\right)=s\mathrm{\Phi }(-u)& \text{(2.4)}\end{array}$$ also is a solution of (2.1). We claim that (2.1) is integrable in terms of the classical hypergeometric functions. At least, this statement is valid under a certain irreducibility condition.

The AKZ is the equation (2.1) with values in the degenerate affine Hecke algebra ${\mathscr{H}}_{A_1}^{\prime }$ of type $A_1$ generated by the elements $s$ and $x$ satisfying the defining relations (2.2) and (2.3): $$\begin{array}{cc}{\mathscr{H}}_{A_1}^{\prime }=⟨s,x⟩/\left\{\text{(2.2), (2.3)}\right\}\text{.}& \text{(2.5)}\end{array}$$ Let $\mathrm{\Phi }\left(u\right)$ be a function of $u$ with values in ${\mathscr{H}}_{A_1}^{\prime }\text{.}$ Note that one can multiply $\mathrm{\Phi }\left(u\right)$ by an arbitrary constant element on the right, i.e. if $\mathrm{\Phi }\left(u\right)$ is a solution, then $\mathrm{\Phi }\left(u\right)a$ $\text{(}a\in {\mathscr{H}}_{A_1}^{\prime }\text{)}$ is also a solution. Let us check the invariance of AKZ (see (2.4)).

We plug in $$\begin{array}{ccc}{\displaystyle -\frac{\partial \stackrel{\sim }{\mathrm{\Phi }}\left(u\right)}{\partial u}}& =& {\displaystyle s(k\frac{s}{e^{-u}-1}+x)\mathrm{\Phi }(-u)}\\ & =& {\displaystyle (k\frac{s}{e^{-u}-1}+sxs)\stackrel{\sim }{\mathrm{\Phi }}\left(u\right)}\end{array}$$ and use $$\frac{1}{1-e^{-u}}=\frac{1}{e^u-1}+1\text{.}$$ Finally, $$\frac{\partial \stackrel{\sim }{\mathrm{\Phi }}\left(u\right)}{\partial u}=(k\frac{s}{e^u-1}+ks-sxs)\stackrel{\sim }{\mathrm{\Phi }}\left(u\right),$$ where $ks-sxs=x\text{.}$

Now we will integrate (2.1). More generally, let us first consider the equation $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial z}=(\frac{A}{1-z}+\frac{B}{z})\mathrm{\Phi }\text{.}}& \text{(2.6)}\end{array}$$ It is (2.1) for $z=e^{-u},$ $A=-ks,$ and $B=-x\text{.}$ The equation (2.6) is much more complicated than the AKZ. However, if $A,B$ are $2\times 2$ matrices acting on the $2\text{-component}$ vector $\mathrm{\Phi },$ this equation is nothing but the hypergeometric differential equation. It readily gives the formulas when $\mathrm{\Phi }$ takes values in irreducible representations of ${\mathscr{H}}_{A_1}^{\prime },$ because the latter exist only in dimensions $1$ or $2\text{.}$

Indeed, a generic $2\text{-dimensional}$ representation $\rho$ of ${\mathscr{H}}_{A_1}^{\prime }$ is given by $$\begin{array}{cc}{\displaystyle \rho \left(s\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\rho \left(x\right)=k(\frac{s}{2}+\left(\begin{array}{cc}0& \zeta \\ \xi & 0\end{array}\right))\text{.}}& \text{(2.7)}\end{array}$$ Because of the gauge transformation $\zeta \to c\zeta ,$ $\xi \to c^{-1}\xi ,$ it is characterized by $\zeta \xi ,$ or by $$\begin{array}{cc}{\displaystyle \mu ={(\zeta \xi +\frac{1}{4})}^{1/2}\text{.}}& \text{(2.8)}\end{array}$$ Then a solution $\mathrm{\Phi }=\left(\genfrac{}{}{0ex}{}{{\mathrm{\Phi }}_1}{{\mathrm{\Phi }}_2}\right)$ for $A=-k\rho \left(s\right),$ $B=-\rho \left(x\right)$ is given in terms of the Gauss hypergeometric function. The first component is $$\begin{array}{cc}{\displaystyle {\mathrm{\Phi }}_1\left(u\right)=z^{-k\mu }{(1-z)}^{k}F(k(1-2\mu ),k,1-2k\mu ;z),}& \text{(2.9)}\end{array}$$ where $z=e^{-u}$ and $F(\alpha ,\beta ,\gamma ;z)=\sum _{n=0}^{\infty }\frac{{\left(\alpha \right)}_n{\left(\beta \right)}_n}{{\left(\gamma \right)}_{n}n!}{z}^n$ with ${\left(x\right)}_n=x(x+1)\cdots (x+n-1)\text{.}$

If $\zeta \xi =0,$ then the representation $\rho$ (2.7) is reducible. In this case, solutions are in terms of elementary functions. We note that the parameters $\alpha ,\beta ,\gamma$ in (2.9) are not arbitrary but obey the constraint $\alpha +1=\beta +\gamma \text{.}$

The AKZ equation for ${GL}_n$

In this section, we introduce the AKZ equation of type ${GL}_n\text{.}$ It can be obtained as a specialization of the standard Knizhnik-Zamolodchikov (KZ) equation from the conformal field theory. The consistency and invariance conditions give rise to the defining relations of the degenerate affine Hecke algebra ${\mathscr{H}}_{{GL}_n}^{\prime }$ introduced by Drinfeld.

Recall that the KZ equation reads $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial z_i}=k\left(\sum _{\underset{j\ne i}{0\le j\le n}}\frac{{\mathrm{\Omega }}_{ij}}{z_i-z_j}\right)\mathrm{\Phi }(0\le i\le n)\text{.}}& \text{(2.10)}\end{array}$$ In the less sophisticated case ${\mathrm{\Omega }}_{ij}$ are the permutation matrices [KZa1984]. Let us assume that ${\mathrm{\Omega }}_{ij}$ are any constant elements (operators) and ${\mathrm{\Omega }}_{ij}={\mathrm{\Omega }}_{ji}\text{.}$ We consider $\mathrm{\Phi }\left(z\right)$ $\text{(}z=(z_0,\dots ,z_n)\text{)}$ taking values in the abstract algebra generated by the elements ${\mathrm{\Omega }}_{ij}\text{.}$ The self-consistency of the system of equations (2.10) means that $$\begin{array}{cc}{\displaystyle \frac{\partial A_j}{\partial z_i}-\frac{\partial A_i}{\partial z_j}=[A_i,A_j],}& \text{(2.11)}\end{array}$$ where $$\begin{array}{cc}{\displaystyle A_i=k\sum _{j\ne i}\frac{{\mathrm{\Omega }}_{ij}}{z_i-z_j}\text{.}}& \text{(2.12)}\end{array}$$ It holds for all values of the complex parameter $k$ if and only if $$\begin{array}{cc}{\displaystyle [{\mathrm{\Omega }}_{ij},{\mathrm{\Omega }}_{kl}]=0,}& \text{(2.13)}\\ {\displaystyle [{\mathrm{\Omega }}_{ij},{\mathrm{\Omega }}_{ik}+{\mathrm{\Omega }}_{jk}]=0,}& \text{(2.14)}\end{array}$$ where the indices $i,j,k,l$ are pairwise distinct. The KZ in this form is due to Aomoto [Aom1988] (it was also studied by Kohno [Koh1987]).

The trigonometric KZ (and the elliptic ones) where introduced for the first time in [Che1989], To be more exact, in this paper the equations which I called the $r\text{-matrix}$ KZ were defined, the connections with Kac-Moody algebras were established, and the reduction procedure was applied to the monodromy (see below). The equations corresponding to the simplest trigonometric $r\text{-matrces}$ (there are many of them due to Belavin and Drinfeld) are closely connected with the AKZ. There were doubts about the importance of my trigonometric KZ in physics when they appeared. Now they are quite common for both mathematicians and physicists.

Consider the group algebra $ℂ\left[{𝕊}_n\right]$ of the permutation group ${𝕊}_n$ of the set $\{1,\dots ,n\}\text{.}$ We denote by $s_{ij}$ the transposition of $i$ and $j\text{.}$ If we set ${\mathrm{\Omega }}_{ij}=s_{ij}$ $\text{(}0\le i,j\le n\text{),}$ the relations (2.13)-(2.14) are satisfied.

Setting $$\begin{array}{cc}{\displaystyle z_0=0,}& \text{(2.15)}\\ {\displaystyle {\mathrm{\Omega }}_{ij}=s_{ij}(i,j\ne 0),}& \text{(2.16)}\\ {\displaystyle {\mathrm{\Omega }}_{0i}=k^{-1}{\mathrm{\Omega }}_i,}& \text{(2.17)}\end{array}$$ the equation (2.10) turns into $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial z_i}=[k\left(\sum _{\underset{j\ne i}{1\le j\le n}}\frac{s_{ij}}{z_i-z_j}\right)+\frac{{\mathrm{\Omega }}_i}{z_i}]\mathrm{\Phi }(1\le i\le n),}& \text{(2.18)}\end{array}$$ and the relations (2.13),(2.14) read as follows: $$\begin{array}{cc}{\displaystyle [s_{ij},{\mathrm{\Omega }}_i+{\mathrm{\Omega }}_j]=0,}& \text{(2.19)}\\ {\displaystyle [k{s}_{ij}+{\mathrm{\Omega }}_i,{\mathrm{\Omega }}_j]=0,}& \text{(2.20)}\\ {\displaystyle [s_{ij},{\mathrm{\Omega }}_l]=0,}& \text{(2.21)}\end{array}$$ where the indices $i,j,l$ are pairwise distinct.

Substituting $$\begin{array}{cc}{\displaystyle z_i=e^{v_i},}& \text{(2.22)}\end{array}$$ we come to $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial v_i}=(k\sum _{j\ne i}\frac{s_{ij}}{1-e^{v_j-v_i}}+{\mathrm{\Omega }}_i)\mathrm{\Phi }\text{.}}& \text{(2.23)}\end{array}$$ Using the elements $$\begin{array}{cc}{\displaystyle y_i={\mathrm{\Omega }}_i+k\sum _{j>i}{s}_{ij},}& \text{(2.24)}\\ {\displaystyle \frac{\partial \mathrm{\Phi }}{\partial v_i}=(k\sum _{j>i}\frac{s_{ij}}{e^{v_i-v_j}-1}-k\sum _{j<i}\frac{s_{ij}}{e^{v_j-v_i}-1}+y_i)\mathrm{\Phi }\text{.}}& \text{(2.25)}\end{array}$$ The elements $\left\{y\right\}$ are convenient since in the limit $$v_1\gg v_2\gg \cdots \gg v_n,$$ we get the system $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial v_i}=y_i\mathrm{\Phi }\text{.}}& \text{(2.26)}\end{array}$$ The consistency of these equations is equivalent to the commutativity $$\begin{array}{cc}{\displaystyle [y_i,y_j]=0\text{.}}& \text{(2.27)}\end{array}$$ We claim that (2.27), a 'limiting self-consistency', together with the relations $$\begin{array}{cc}{\displaystyle [s_i,y_j]=0\text{if}\hspace{0.17em}j\ne i,i+1,}& \text{(2.28)}\\ s_i{y}_i-y_{i+1}{s}_i=k,& \text{(2.29)}\end{array}$$ where $s_i=s_{i\hspace{0.17em}i+1}$ $\text{(}1\le i\le n-1\text{),}$ ensure (2.19)-(2.21).

It can be put in the following way. Let us introduce the degenerate affine Heche algebra of type ${GL}_n$ as an algebraic span of $ℂ\left[{𝕊}_n\right]$ and $y_i$ $\text{(}1\le i\le n\text{)}$ with the relations (2.27), (2.28) and (2.29), denoting it by ${\mathscr{H}}_{{GL}_n}^{\prime },$ or simply by ${\mathscr{H}}_n^{\prime }\text{.}$ We call the system (2.25) with the values in ${\mathscr{H}}_n^{\prime }$ the AKZ of type ${GL}_n\text{.}$ It is well-defined, i.e. self-consistent.

The ${𝕊}_n\text{-invariance}$

In this section, we discuss the ${𝕊}_n\text{-symmetry}$ of the AKZ equation introduced in the previous section and clarify in full the definition of ${\mathscr{H}}_n^{\prime }\text{.}$

The group ${𝕊}_n$ acts on ${ℂ}^n$ naturally by $$v=(v_1,\dots ,v_n)\in {ℂ}^n\mapsto w\left(v\right)=(v_{i_1},\dots ,v_{i_n})\in {ℂ}^n$$ for $w^{-1}=(i_1,i_2,\dots ,i_n)\in {𝕊}_n\text{.}$ Given a function $\mathrm{\Phi }\left(v\right)$ of $v\in {ℂ}^n$ with values in ${\mathscr{H}}_n^{\prime },$ we define the action of $w\in ℂ\left[{𝕊}_n\right]$ on $\mathrm{\Phi }\left(v\right)$ by $$\begin{array}{cc}\left(w\left(\mathrm{\Phi }\right)\right)\left(v\right)=w·\mathrm{\Phi }\left(w^{-1}\left(v\right)\right)\text{.}& \text{(2.30)}\end{array}$$ Here the dot means the product in ${\mathscr{H}}_n^{\prime }\text{.}$ It follows from (2.28) and (2.29) that if $\mathrm{\Phi }$ solves (2.25), then so does $w\left(\mathrm{\Phi }\right)\text{.}$ Just conjugate the equations by $\left\{s_i\right\}\text{.}$ Moreover, the invariance is exactly equivalent to the relations (2.28) and (2.29). Thus the invariance and the limiting self-consistency (2.27) give the self-consistency of our system for all $k\text{.}$

Degenerate affine Hecke algebras

In this section, we fix notations for root systems and define the degenerate affine Hecke algebra for an arbitrary root system.

Let $\mathrm{\Sigma }$ be a root system in ${ℝ}^n$ with the inner product $(\hspace{0.17em},\hspace{0.17em})\text{.}$ Choose a system of simple roots ${\alpha }_1,\dots ,{\alpha }_n$ of $\mathrm{\Sigma }$ and denote by ${\mathrm{\Sigma }}_{+}$ the set of positive roots. For a root $\alpha \in \mathrm{\Sigma },$ define the coroot ${\alpha }^{\vee }$ by $${\alpha }^{\vee }=\frac{2\alpha }{(\alpha ,\alpha )}$$ and the reflection $s_{\alpha }$ by $$s_{\alpha }\left(u\right)=u-({\alpha }^{\vee },u)\alpha (u\in {ℝ}^n)\text{.}$$ We will denote $s_{{\alpha }_i}$ simply by $s_i\text{.}$ The fundamental coweights $b_i$ are as follows: $$(b_i,{\alpha }_j)={\delta }_{ij}\text{.}$$ We also use the notation $a_i={\alpha }_i^{\vee }\text{.}$ For $u\in {ℝ}^n,$ the coordinates will be $u_i=(u,{\alpha }_i)\text{.}$ We also set $u_{\alpha }=(u,\alpha )$ for $\alpha \in \mathrm{\Sigma }\text{.}$ Check that $$\frac{\partial u_{\alpha }}{\partial u_i}={\nu }_{\alpha }^i=(b_i,\alpha )=\text{the multiplicity of}\hspace{0.17em}{\alpha }_i\hspace{0.17em}\text{in}\hspace{0.17em}\alpha \text{.}$$

Let $W$ be the Weyl group of $\mathrm{\Sigma }\text{:}$ $W=⟨s_{\alpha },\alpha \in \mathrm{\Sigma }⟩=⟨s_1,\dots ,s_n⟩\text{.}$ Define the action of $W$ on functions on ${ℝ}^n$ by $$\begin{array}{cc}{}^{w}f\left(u\right)=j\left(w^{-1}\left(u\right)\right)(u\in {ℝ}^n)\text{.}& \text{(2.31)}\end{array}$$

Then we have $${}^w{u}_{\alpha }=(w^{-1}\left(u\right),\alpha )=u_{w\left(\alpha \right)}\text{.}$$

Now we can define the degenerate affine Hecke algebra ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ associated with $\mathrm{\Sigma }\text{.}$ This definition is due to Lusztig [Lus1989] (he calls it the graded affine Hecke algebra, considering $k$ as a formal parameter). Drinfeld introduced this algebra in the ${GL}_n\text{-case}$ in [Dri1986-2] prior to Lusztig. These algebras are natural degenerations of the corresponding $p\text{-adic}$ ones.

Definition 2.1. Let ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$ be the associative algebra generated by $ℂ\left[W\right]$ and $x_1,\dots ,x_n$ with the following relations $$\begin{array}{cc}[x_i,x_j]=0,\forall i,j,& \text{(2.32)}\\ [s_i,x_j]=0,\text{if}i\ne j,& \text{(2.33)}\\ s_i{x}_i-{\hat{x}}_i{s}_i=k\text{.}& \text{(2.34)}\end{array}$$ Here $k$ is a complex number and $$\begin{array}{cc}{\displaystyle {\hat{x}}_i=x_i-\sum _{j=1}^n({\alpha }_i^{\vee },{\alpha }_j)x_j\text{.}}& \text{(2.35)}\end{array}$$

Introducing $$\begin{array}{cc}{\displaystyle x_b=\sum _{i=1}^n(b,{\alpha }_i)x_i=\sum _{i=1}^n{k}_i{x}_i\text{for}b=\sum _{i=1}^n{k}_i{b}_i,}& \text{(2.36)}\end{array}$$ we can express the right hand side of (2.35) as $x_{s_i\left(b_i\right)}=x_{b_i}-x_{a_i}\text{.}$ More generally, $$\begin{array}{cc}{s}_i{x}_b-x_{s_i\left(b\right)}{s}_i=x_b{s}_i-s_i{x}_{s_i\left(b\right)}=k(b,{\alpha }_i)\text{.}& \text{(2.37)}\end{array}$$

Later we will use the following partial derivatives $$\begin{array}{cc}{\partial }_b\left(u_{\alpha }\right)=(\alpha ,b)\text{.}& \text{(2.38)}\end{array}$$ For instance, $$\frac{\partial }{\partial u_i}={\partial }_i={\partial }_{b_i}$$

The AKZ equation associated with ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }$

In this section we introduce the AKZ equation associated with the root system $\mathrm{\Sigma },$ and give several examples.

Let us consider the following system of partial differential equations $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u_i}=(k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{\nu }_{\alpha }^i\frac{s_{\alpha }}{e^{u_{\alpha }}-1}+x_i)\mathrm{\Phi }(1\le i\le n)\text{.}}& \text{(2.39)}\end{array}$$ Here $k$ is a complex number. We denote the right hand side of (2.39) as $A_i\mathrm{\Phi }\text{.}$ First we assume that $\mathrm{\Phi }$ takes values in an associative algebra generated by $ℂ\left[W\right]$ and $x_1,\dots ,x_n\text{.}$ We say that the system (2.39) is self-consistent provided that $$\begin{array}{cc}{\displaystyle [\frac{\partial }{\partial u_i}-A_i,\frac{\partial }{\partial u_j}-A_j]=0\text{.}}& \text{(2.40)}\end{array}$$ It is called invariant if, for any solution $\mathrm{\Phi }$ of (2.39) and any element $w$ of $W,$ $w\left(\mathrm{\Phi }\right)$ (see 2.30) is again a solution of (2.39).

Theorem 2.1. The system (2.39) is self-consistent and invariant if and only if $s_1,\dots ,s_n,$ $x_1,\dots ,x_n$ satisfy the relations (2.32), (2.33), (2.34) defining ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$

We introduce the AKZ equation associated with $\mathrm{\Sigma }$ to be the system (2.39) for functions $\mathrm{\Phi }$ with values in ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime }\text{.}$

Using the notation $x_b$ and ${\partial }_b$ from (2.36) and (2.38), the system (2.39) can be expressed as $$\begin{array}{cc}{\displaystyle {\partial }_b\mathrm{\Phi }=(k\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}(b,\alpha )\frac{s_{\alpha }}{e^{u_{\alpha }}-1}+x_b)\mathrm{\Phi }\text{.}}& \text{(2.41)}\end{array}$$

Remark 2.1. The parameter $k$ may depend on the lengths of roots. Generally speaking the AKZ equation is as follows: $$\begin{array}{cc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u_i}=(\sum _{\alpha \in {\mathrm{\Sigma }}_{+}}{k}_{\left|\alpha \right|}{\nu }_{\alpha }^i\frac{s_{\alpha }}{e^{u_{\alpha }}-1}+x_i)\mathrm{\Phi }\text{.}}& \text{(2.42)}\end{array}$$

Let us write down the explicit forms of the AKZ equation in the simplest cases.

Example 2.1. When $\mathrm{\Sigma }=A_1,$ the AKZ equation is exactly (2.1).

Example 2.2. For $A_2,$ the AKZ equation is $$\begin{array}{ccc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u_1}}& =& {\displaystyle \{k(\frac{s_{12}}{e^{u_1}-1}+\frac{s_{13}}{e^{u_1+u_2}-1})+x_1\}\mathrm{\Phi },}\\ {\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u_2}}& =& {\displaystyle \{k(\frac{s_{23}}{e^{u_2}-1}+\frac{s_{13}}{e^{u_1+u_2}-1})+x_2\}\mathrm{\Phi },}\end{array}$$ where $s_{ij}$ denotes the transposition of $i$ and $j\text{.}$ In this case ${\hat{x}}_1=x_2-x_1,$ ${\hat{x}}_2=x_1-x_2\text{.}$

Example 2.3. The root system $B_2$ is realized in the following way. Let ${ϵ}_1$ and ${ϵ}_2$ form an orthonormal basis of ${ℝ}^2\text{.}$ Then the set of positive roots consists of the following vectors: $$\begin{array}{c}{\alpha }_1={ϵ}_1-{ϵ}_2,\\ {\alpha }_2={ϵ}_2,\\ {\alpha }_1+{\alpha }_2={ϵ}_1,\\ {\alpha }_1+2{\alpha }_2={ϵ}_1+{ϵ}_2\text{.}\end{array}$$ Let $s=s_1$ and $t=s_2\text{.}$ Then $s$ and $t$ satisfy $tsts=stst$ (the Coxeter relation for $W_{B_2}=W_{C_2}\text{)}$ and $s^2=1,$ $t^2=1\text{.}$ In this case ${\hat{x}}_1=x_2-x_1,$ ${\hat{x}}_2=2x_1-x_2\text{.}$ The AKZ equation reads as follows: $$\begin{array}{ccc}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u_1}}& =& {\displaystyle \{k(\frac{s}{e^{u_1}-1}+\frac{sts}{e^{u_1+u_2}-1}+\frac{tst}{e^{u_1+2u_2}-1})+x_1\}\mathrm{\Phi }}\\ {\displaystyle \frac{\partial \mathrm{\Phi }}{\partial u_2}}& =& {\displaystyle \{k(\frac{s}{e^{u_2}-1}+\frac{sts}{e^{u_1+u_2}-1}+\frac{tst}{e^{u_1+2u_2}-1})+x_2\}\mathrm{\Phi }\text{.}}\end{array}$$ Note the appearance of the coefficient 2 in the latter. In the case of $E_8$ the coefficients are from $1$ to $6$ (otherwise they are less than $6\text{).}$

The $A_{n-1}$ case

In this section, we will show that the AKZ equation of type ${GL}_n$ discussed in §2.2 reduces to the AKZ equation for the root system $\mathrm{\Sigma }\subset {ℝ}^{n-1}$ of type $A_{n-1}\text{.}$

First note that $$\begin{array}{cc}x=y_1+\dots +y_n& \text{(2.43)}\end{array}$$ is central in the algebra ${\mathscr{H}}_n^{\prime }\text{.}$ By setting $$\begin{array}{cc}{\displaystyle x_i=y_1+\dots +y_i-\frac{i}{n}x,}& \text{(2.44)}\end{array}$$ we have an embedding of ${\mathscr{H}}_{\mathrm{\Sigma }}^{\prime },$ where $\mathrm{\Sigma }$ is the root system of type $A_{n-1},$ into ${\mathscr{H}}_n^{\prime }\text{.}$ We put $$\begin{array}{cc}{u}_i=v_i-v_{i+1}(1\le i\le n-1)\text{.}& \text{(2.45)}\end{array}$$ The space ${ℝ}^{n-1}$ is identified with the quotient space of ${ℝ}^n=\left\{\sum _{i=1}^n{v}_i{ϵ}_i\hspace{0.17em}\right|\hspace{0.17em}{v}_i\in ℝ\}$ $${ℝ}^{n-1}\simeq {\oplus }_{i=1}^nℝ{ϵ}_i/ℝϵ$$ where ${\left\{{ϵ}_i\right\}}_{1\le i\le n}$ is the orthonormal basis and $ϵ={ϵ}_1+\dots +{ϵ}_n\text{.}$ From (2.25) we have $$\sum _{i=1}^n\frac{\partial \mathrm{\Phi }}{\partial v_i}=x\mathrm{\Phi }\text{.}$$ Therefore, the function $$\mathrm{\Phi }\prime \left(v\right)=e^{-x·\frac{1}{n}(v_1+\cdots +v_n)}\mathrm{\Phi }\left(v\right)$$ is well-defined on the quotient space ${ℝ}^{n-1}\text{.}$ Now it is straightforward to see that (2.25) reduces to the AKZ equation of type $A_{n-1}$ for $\mathrm{\Phi }\prime \left(v\right)$ $$\frac{\partial \mathrm{\Phi }\prime }{\partial u_i}=(k\sum _{j\le i<l}\frac{s_{jl}}{e^{u_j+\cdots +u_{i-1}}-1}+x_i)\mathrm{\Phi }\prime (1\le i\le n-1)\text{.}$$

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