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: 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.

Double affine Hecke algebras and Macdonald polynomials

Macdonald polynomials : the $A_1$ case

The subject of this section is to show how the Hecke algebra technique is applied to the Macdonald polynomials. We will concentrate on the duality and the recurrence relations. The key notion will be the double affine Hecke. Let us start with $A_1\text{.}$

The corresponding $L\text{-operator}$ in the differential case reads as follows $$\begin{array}{cc}{\displaystyle L^{\left(k\right)}=\frac{{\partial }^2}{\partial u^2}+2k\frac{e^u+e^{-u}}{e^u-e^{-u}}\frac{\partial }{\partial u}+k^2,}& \text{(5.1)}\end{array}$$ where $k$ is a complex parameter. There are two special values of $k$ when the operator $L^{\left(k\right)}$ is very simple. For $k=0$ we have $L^{\left(0\right)}={\partial }^2/\partial u^2\text{.}$ When $k=1,$ $$\begin{array}{cc}{\displaystyle L^{\left(1\right)}=d^{-1}\frac{{\partial }^2}{\partial u^2}d,\text{with}d=e^u-e^{-u}\text{.}}& \text{(5.2)}\end{array}$$ Similarly, we can conjugate by $d^k$ for any $k\text{:}$ $$\begin{array}{cc}{\displaystyle d^k{L}^{\left(k\right)}{d}^{-k}=\frac{{\partial }^2}{\partial u^2}-\frac{4k(k-1)}{{(e^u-e^{-u})}^2}\text{.}}& \text{(5.3)}\end{array}$$ Sometimes $L$ is more convenient to deal with in this form.

Let us now consider the eigenvalue problem for the operator $L^{\left(k\right)}\text{:}$ $$\begin{array}{cc}{L}^{\left(k\right)}\phi ={\lambda }^2\phi \text{.}& \text{(5.4)}\end{array}$$ If $k=1,$ the solution of this equation is immediate: $$\begin{array}{cc}{\displaystyle \phi (u;\lambda )=\frac{\text{sinh}\left(u\lambda \right)}{\text{sinh}\left(u\right)\text{sinh}\left(\lambda \right)}\text{.}}& \text{(5.5)}\end{array}$$ In this normalization it is symmetric with respect to $u$ and $\lambda \text{.}$ Without $\text{sinh}\left(\lambda \right)$ it generalizes the characters of finite-dimensional representations of ${SL}_2\left(ℂ\right)$ $\text{(}k=1\text{).}$

If $k=1/2,$ this operator is the radial part of the Casimir operator for the symmetric space ${SL}_2\left(ℝ\right)/{SO}_2\left(ℝ\right)\text{.}$ It is the restriction of the Casimir operator $C$ on the double coset space ${SO}_2\backslash {SL}_2/{SO}_2$ which is identified with a domain in ${ℝ}^{*}/S_2\text{.}$ If $k=1,2,$ then $L^{\left(k\right)}$ corresponds in the same way to ${SL}_2\left(ℂ\right)/SU\left(2\right)$ and ${SL}_2\left(𝒦\right)/{SU}_2\left(𝒦\right)$ for the quaternions $𝒦\text{.}$

For any $k,$ one can find a family of even $\text{(}u\to -u\text{)}$ solutions of the form $$\begin{array}{cc}{p}_n=e^{nu}+e^{-nu}+\text{lower integral exponents,}& \text{(5.6)}\end{array}$$ such that $$\begin{array}{cc}{L}^{\left(k\right)}{p}_n={(n+k)}^2{p}_n& \text{(5.7)}\end{array}$$ for $n=0,1,2,\dots \text{.}$ This family of hyperbolic polynomials satisfy the orthogonality relations $$\begin{array}{cc}\text{Constant Term}\hspace{0.17em}\left(p_n{p}_m{d}^{2k}\right)=c_n{\delta }_{nm}\text{.}& \text{(5.8)}\end{array}$$ They are called the ultraspherical polynomials.

We can also consider the rational limit $$\begin{array}{cc}{\displaystyle {\ell }^{\left(k\right)}=\frac{{\partial }^2}{\partial u^2}+\frac{2k}{u}\frac{\partial }{\partial u}}& \text{(5.9)}\end{array}$$ of the operator $L^{\left(k\right)},$ switching from the Sutherland model to the Calogero model. The solutions of the rational eigenvalue problem are expressed in terms of the Bessel function. In this case, the solutions can be normalized to ensure the symmetry between the variable and the eigenvalue. In the trigonometric case it is possible only for two special values $k=0,1\text{.}$ It is one of the main demerits of the harmonic analysis on the symmetric spaces.

In the difference theory, this symmetry holds for any root systems. This discovery is expected to renew the Harish-Chandra theory. The so-called group case $\text{(}k=1\text{)}$ is an intersection point of the differential (classical) and difference (new) theories.

We now turn to the difference version. We set $x=e^u$ and introduce the 'multiplicative difference' ${\mathrm{\Gamma }}_q$ acting as ${\mathrm{\Gamma }}_q\left(f\left(x\right)\right)=f\left(qx\right)$ and satisfying the commutation relation ${\mathrm{\Gamma }}_{q}x=qx{\mathrm{\Gamma }}_q\text{.}$ The Macdonald operator $L$ is expressed as follows: $$\begin{array}{cc}{\displaystyle L=\frac{tx-t^{-1}{x}^{-1}}{x-x^{-1}}{\mathrm{\Gamma }}_q+\frac{t{x}^{-1}-t^{-1}x}{x^{-1}-x}{\mathrm{\Gamma }}_q^{-1}\text{.}}& \text{(5.10)}\end{array}$$

The parameter $k$ in the difference setup is determined from the relation $t=q^k\text{.}$ When $q=t$ (or $k=1\text{),}$ the operator $L$ is simple: $$\begin{array}{cc}{\displaystyle L=\frac{1}{x-x^{-1}}({\mathrm{\Gamma }}_q+{\mathrm{\Gamma }}_q^{-1})(x-x^{-1})\text{.}}& \text{(5.11)}\end{array}$$ Compare this formula with (5.2) in the differential case and notice that (5.11) is easier to check than (5.2).

The eigenvalue problem $$\begin{array}{cc}L\phi =(\mathrm{\Lambda }+{\mathrm{\Lambda }}^{-1})\phi & \text{(5.12)}\end{array}$$ always has a self-dual family of solutions. When $n=0,1,2,\dots ,$ there exists a unique family of the so-called $q,$ $t\text{-ultraspherical}$ (or Roger-Askey-Ismail) Laurent polynomials $$\begin{array}{cc}{p}_n=x^n+x^{-n}+\text{lower terms,}& \text{(5.13)}\end{array}$$ which are symmetric with respect to the transformation $x\to x^{-1},$ and satisfy the equation $$\begin{array}{cc}L{p}_n=(t{q}^n+t^{-1}{q}^{-n})p_n\text{.}& \text{(5.14)}\end{array}$$ The following duality theorem is proved in the next section by using the double affine Hecke algebra.

Theorem 5.1 (Duality). $p_n\left(t{q}^m\right)p_m\left(t\right)=p_m\left(t{q}^n\right)p_n\left(t\right)$ for any $m,n=0,1,2,\dots \text{.}$

If we set $$\begin{array}{cc}{\displaystyle {\pi }_n\left(x\right)=\frac{p_n\left(x\right)}{p_n\left(t\right)},}& \text{(5.15)}\end{array}$$ the duality can be rewritten as follows: $$\begin{array}{cc}{\pi }_n\left(t{q}^m\right)={\pi }_m\left(t{q}^n\right)\text{(}m,n=0,1,2,\dots \text{).}& \text{(5.16)}\end{array}$$

The Askey-Ismail polynomials are nothing but the Macdonald polynomials of type $A_1\text{.}$ There are three main Macdonald's conjectures for the Macdonald polynomials associated with root systems (see [Mac0011046, Mac1423624]):

(1)	the scalar product conjecture,
(2)	the evaluation conjecture,
(3)	the duality conjecture.

One may also add the Pieri rules to the list. These conjectures were justified recently using the double affine Hecke algebras in [Che1993, Che1995-2].

A modern approach to $q,$ $t\text{-ultraspherical}$ polynomials

The duality from Theorem 5.1 can be rephrased as the symmetry of a certain scalar product. Actually this product is a difference counterpart of the spherical Fourier transform. For any symmetric Laurent polynomials $f,g\in ℂ[x+x^{-1}],$ we set $$\begin{array}{cc}\{f,g\}=\left(a\left(L\right)g\right)\left(t\right),& \text{(5.17)}\end{array}$$ where $a$ is a polynomial such that $f\left(x\right)=a(x+x^{-1})\text{.}$ So we apply the operator $a\left(L\right)$ to $g$ and then evaluate the result at $x=t\text{.}$

Theorem 5.2. $\{f,g\}=\{g,f\}$ for any $f,g\in ℂ[x+x^{-1}]\text{.}$

Theorem 5.1 follows from Theorem 5.2. Indeed, if $f={\pi }_m$ and $g={\pi }_n,$ we can compute the scalar product as follows: $$\begin{array}{cc}\{{\pi }_m,{\pi }_n\}=\left(a\left(L\right){\pi }_n\right)\left(t\right)=a(t{q}^n+t^{-1}{q}^{-n}){\pi }_n\left(t\right)={\pi }_m\left(q{t}^n\right)\text{.}& \text{(5.18)}\end{array}$$ Use $L{\pi }_n=(t{q}^n+t^{-1}{q}^{-n}){\pi }_n$ and the normalization ${\pi }_n\left(t\right)=1\text{.}$ Hence Theorem 5.2 implies ${\pi }_m\left(q{t}^n\right)={\pi }_n\left(q{t}^m\right)\text{.}$ Actually the theorems are equivalent, since $p_n$ form a basis in the space of all symmetric Laurent polynomials.

Definition 5.1. The double affine Hecke algebra ${\mathscr{H}}^{q,t}$ of type $A_1$ is the quotient $$\begin{array}{cc}{\mathscr{H}}^{q,t}=⟨X,Y,T⟩/\sim ,& \text{(5.19)}\end{array}$$ by the relations for the generators $X,Y,T$ $$\begin{array}{cc}TXT=X^{-1},T^{-1}Y{T}^{-1}=Y^{-1},& \text{(5.20)}\\ Y^{-1}{X}^{-1}YX{T}^2=q^{-1},(T-t)(T+t^{-1})=0\text{.}& \text{(5.21)}\end{array}$$

Here we consider $q,t$ as numbers or parameters. The first point of the theory is the following statement of PBW type.

Any element of $H\in \mathscr{H}$ can be uniquely expressed in the form $$\begin{array}{cc}{\displaystyle H=\sum _{\underset{e=0,1}{i,j\in \mathbb{Z}}}{c}_{iej}{X}^i{T}^e{Y}^j(c_{iej}\in ℂ)\text{.}}& \text{(5.22)}\end{array}$$ The second important fact is the symmetry of ${\mathscr{H}}^{q,t}$ with respect to $X$ and $Y\text{.}$

Theorem 5.3. There exists an anti-involution $\varphi :\mathscr{H}\to \mathscr{H}$ such that $\varphi \left(X\right)=Y^{-1},$ $\varphi \left(Y\right)=X^{-1}$ and $\varphi \left(T\right)=T\text{.}$

Indeed, $\varphi$ transposes the first two relations and leaves the remaining invariant.

Next, we introduce the expectation value ${\left\{H\right\}}_0\in ℂ$ of an element $H\in {\mathscr{H}}^{q,t}$ by $$\begin{array}{cc}{\displaystyle {\left\{H\right\}}_0=\sum _{\underset{e=0,1}{i,j\in \mathbb{Z}}}{c}_{iej}{t}^{-i}{t}^e{t}^j,}& \text{(5.23)}\end{array}$$ using the expression (5.22). The definitions of $\varphi$ and ${\left\{\hspace{0.17em}\right\}}_0$ give that $$\begin{array}{cc}{\left\{\varphi \left(H\right)\right\}}_0={\left\{H\right\}}_0\text{for any}\hspace{0.17em}H\in {\mathscr{H}}^{q,t}\text{.}& \text{(5.24)}\end{array}$$ Now we can introduce the operator counterpart of the pairing $\{f,g\}=\{g,f\}$ on ${\mathscr{H}}^{q,t}\times {\mathscr{H}}^{q,t}\text{:}$ setting $$\begin{array}{cc}{\{A,B\}}_0={\left\{\varphi \left(A\right)B\right\}}_0& \text{(5.25)}\end{array}$$ for any $A,B\in {\mathscr{H}}^{q,t}\text{.}$

The $\varphi \text{-invariance}$ of the expectation value (5.24) ensures that it is symmetric $${\{A,B\}}_0={\{B,A\}}_0\text{.}$$ We also remark that this pairing is non-degenerate for generic $q,t\text{.}$

Theorem 5.2 readily follows from

Lemma 5.4. For any symmetric Laurent polynomials $f\left(x\right),g\left(x\right)\in ℂ[x+x^{-1}],$ $${\{f\left(X\right),g\left(X\right)\}}_0=\{f,g\}\text{.}$$

To prove the lemma we need to introduce the basic representation of the double affine Hecke algebra ${\mathscr{H}}^{q,t}\text{.}$ Consider the one-dimensional representation of the Hecke algebra ${\mathscr{H}}_Y=⟨T,Y⟩$ sending $T\mapsto t$ and $Y\mapsto t\text{.}$ We denote this representation simply by $+\text{.}$ Then take the induced representation $$\begin{array}{cc}V={\text{Ind}}_{{\mathscr{H}}_Y}^{\mathscr{H}}(+)=\mathscr{H}/\{\mathscr{H}(T-t)+\mathscr{H}(Y-t)\}\simeq ℂ[x,x^{-1}],& \text{(5.26)}\end{array}$$ where the last isomorphism is $x^n\leftrightarrow X^n$ mod $\mathscr{H}(T-t)+\mathscr{H}(Y-t)\text{.}$ Under this identification of $V$ with the ring $ℂ[x,x^{-1}]$ of Laurent polynomials, the element $X$ acts on $ℂ[x,x^{-1}]$ as the multiplication by $x,$ while $T$ and $Y$ act by the operators $$\begin{array}{cc}{\displaystyle \hat{T}=ts+\frac{t-t^{-1}}{x^2-1}(s-1)\text{and}\hat{Y}=s{\mathrm{\Gamma }}_q\hat{T},}& \text{(5.27)}\end{array}$$ respectively. Here $s\left(f\right)\left(x\right)=f\left(x^{-1}\right),$ the equality $Hf\left(x\right)=g\left(x\right)$ in $V$ means that $Hf\left(X\right)-g\left(X\right)\in \mathscr{H}(T-t)+\mathscr{H}(Y-t)\text{.}$ The latter readily gives the desired formulas for $\hat{T},\hat{Y}\text{.}$

The expectation value is the composition $$\begin{array}{cc}\mathscr{H}\stackrel{\alpha }{⟶}C\cong ℂ[x,x^{-1}]\stackrel{\beta }{⟶}ℂ,& \text{(5.28)}\end{array}$$ where $\alpha$ is a residue mod $\mathscr{H}(T-t)+\mathscr{H}(Y-t)$ and $\beta \left(f\right)=f\left(t^{-1}\right)$ is the evaluation map at $t^{-1}\text{.}$ Take any $f,g\in ℂ[X,X^{-1}]\text{.}$ Then $$\begin{array}{cc}{\{f\left(X\right),g\left(X\right)\}}_0={\left\{\varphi \left(f\left(X\right)\right)g\left(X\right)\right\}}_0={\left\{f\left(Y^{-1}\right)g\left(X\right)\right\}}_0=f\left({\hat{Y}}^{-1}\right)\left(g\right)\left(t^{-1}\right)\text{.}& \text{(5.29)}\end{array}$$ The last equality follows from (5.28). If $f$ and $g$ are symmetric and $f\left(X\right)=a(X+X^{-1}),$ then $$\begin{array}{cc}{\{f\left(X\right),g\left(X\right)\}}_0=a\left(L\right)\left(g\right)\left(t\right)=\{f,g\},& \text{(5.30)}\end{array}$$ since the operator $\hat{Y}+{\hat{Y}}^{-1}$ acts on symmetric Laurent polynomials as $L\text{.}$ It is straightforward. The duality is established.

This method of proving of the duality theorem can be generalized to any root system.

We now discuss the application of the duality to the Pieri rules, the recurrence formulas for ${\pi }_n\text{'s}$ with respect to the index $n\text{.}$ First we will discretize functions and operators.

Recall that the renormalized $q,t\text{-ultraspherical}$ polynomials ${\pi }_n\left(x\right)$ are characterized by the conditions $$\begin{array}{cc}L{\pi }_n=(t{q}^n+t^{-1}{q}^{-n}){\pi }_n,{\pi }_n\left(t\right)=1,& \text{(5.31)}\end{array}$$ where $$\begin{array}{cc}{\displaystyle L=\frac{tx-t^{-1}{x}^{-1}}{x-x^{-1}}\mathrm{\Gamma }+\frac{t{x}^{-1}-t^{-1}x}{x^{-1}-x}{\mathrm{\Gamma }}^{-1}\text{.}}& \text{(5.32)}\end{array}$$ As always, $\mathrm{\Gamma }x=qx\mathrm{\Gamma }\text{.}$ Denote the set of $ℂ\text{-valued}$ functions on $\mathbb{Z}$ by $\text{Funct}(\mathbb{Z},ℂ)\text{.}$ For any Laurent polynomial $f\in ℂ[x,x^{-1}]$ or more general rational function, we define $\hat{f}\in \text{Funct}(\mathbb{Z},ℂ)$ by setting $$\begin{array}{cc}\hat{f}\left(m\right)=f\left(t{q}^m\right)\text{for all}m\in \mathbb{Z}\text{.}& \text{(5.33)}\end{array}$$ Considering $𝒜=⟨ℂ\left(x\right),\mathrm{\Gamma }⟩$ as an abstract algebra with the fundamental relation $\mathrm{\Gamma }x=qx\mathrm{\Gamma },$ the action of $𝒜$ on $\phi \in \text{Funct}(\mathbb{Z},ℂ)$ is as follows: $$\begin{array}{cc}\hat{x}\phi \left(m\right)=t{q}^m\phi \left(m\right),\hat{\mathrm{\Gamma }}\phi \left(m\right)=\phi (m+1)\text{.}& \text{(5.34)}\end{array}$$ The correspondence $f\mapsto \hat{f},$ whenever it is well-defined (the functions $f$ may have denominators), is an $𝒜\text{-homomorphism}$ $ℂ\left(x\right)\to \text{Funct}(\mathbb{Z},ℂ)\text{.}$

Due to (5.31): $$\begin{array}{cc}\hat{L}{\hat{\pi }}_n\left(m\right)=(t{q}^{-n}+t^{-1}{q}^n){\hat{\pi }}_n\left(m\right)\text{.}& \text{(5.35)}\end{array}$$ The Pieri rules result directly from this equality. Indeed, the duality ${\hat{\pi }}_n\left(m\right)={\hat{\pi }}_m\left(n\right)$ implies: $$\begin{array}{cc}ℒ{\hat{\pi }}_m\left(n\right)=(t{q}^n+t^{-1}{q}^{-n}){\hat{\pi }}_m\left(n\right)=(\hat{x}+{\hat{x}}^{-1}){\hat{\pi }}_m\left(n\right)\text{.}& \text{(5.36)}\end{array}$$ Here $ℒ$ is $\hat{L}$ acting on the indices $m$ instead of $n\text{.}$ Explicitly, $$\begin{array}{cc}{\displaystyle \frac{t^2{q}^m-t^{-2}{q}^{-m}}{t{q}^m-t^{-1}{q}^{-m}}{\hat{\pi }}_{m-1}\left(n\right)+\frac{q^{-m}-q^m}{t^{-1}{q}^{-m}-t{q}^m}{\hat{\pi }}_{m-1}\left(n\right)=(\hat{x}+{\hat{x}}^{-1}){\hat{\pi }}_m\left(n\right)\text{.}}& \text{(5.37)}\end{array}$$ For generic $q,t,$ the mapping $f\to \hat{f}$ is injective. Hence one can pull (5.37) back, removing the hats: $$\begin{array}{cc}{\displaystyle (x+x^{-1}){\pi }_m=\frac{t^2{q}^m-t^{-2}{q}^{-m}}{t{q}^m-t^{-1}{q}^{-m}}{\pi }_{m+1}+\frac{q^m-q^{-m}}{t{q}^m-t^{-1}{q}^{-m}}{\pi }_{m-1}\text{.}}& \text{(5.38)}\end{array}$$ This is the Pieri formula in the case of $A_1\text{.}$ See [AIs1983]. We remark that this formula makes sense when $m=0,$ since the coefficient of ${\pi }_{m-1}$ vanishes at $m=0\text{.}$ Generally speaking the 'vanishing conditions' are much less obvious.

The Pieri rules obtained above can be used to prove the so-called evaluation conjecture describing the values of $p_n$ at $x=t\text{.}$ Applying (5.38) repeatedly, we get the formula $$\begin{array}{cc}{(x+x^{-1})}^{\ell }{\pi }_m=c_{\ell ,m}{\pi }_{m+\ell }+\text{lower terms}& \text{(5.39)}\end{array}$$ for each $\ell =0,1,2,\dots \text{.}$ The leading coefficient $c_{\ell ,m}$ can be readily calculated: $$\begin{array}{cc}{\displaystyle c_{\ell ,m}=\prod _{i=0}^{\ell -1}\frac{t^2{q}^{m+i}-t^{-2}{q}^{-m-i}}{t{q}^{m+i}-t^{-1}{q}^{-m-i}}\text{.}}& \text{(5.40)}\end{array}$$ Let us look at (5.39) for $m=0\text{:}$ $$\begin{array}{cc}{(x+x^{-1})}^{\ell }=c_{\ell ,0}{\pi }_{\ell }+\text{lower terms.}& \text{(5.41)}\end{array}$$ Comparing the coefficients of $x^l+x^{-l},$ we have $1=c_{\ell ,0}/p_{\ell }\left(t\right),$ since $p_{\ell }=x^{\ell }+x^{-\ell }+\cdots \text{.}$ Hence $$\begin{array}{cc}{\displaystyle p_{\ell }\left(t\right)=c_{\ell ,0}=\prod _{i=0}^{\ell -1}\frac{t^2{q}^i-t^{-2}{q}^{-i}}{t{q}^i-t^{-1}{q}^{-i}}\text{.}}& \text{(5.42)}\end{array}$$

This value is easy to calculate directly (the formulas for $p_n$ are known). However the method described in this section is applicable to arbitrary root systems. We need only the duality, which is the main advantage of the difference harmonic analysis in contrast to the classical Harish-Chandra theory.

The ${GL}_n$ case

In this last subsection, we will discuss the double affine Hecke algebra and applications for ${GL}_n\text{.}$ Since we have already clarified the $A_1$ case in full detail, we will try to get concentrated on the main points only.

In the ${GL}_n$ case, the Macdonald operators $M_0=1,$ $M_1,\dots ,M_n$ are as follows: $$\begin{array}{cc}{\displaystyle M_m=\sum _{I(i_1<\dots <i_m)}\prod _{\underset{j\notin I}{i\in I}}\frac{t{x}_i-t^{-1}{x}_j}{x_i-x_j}{\mathrm{\Gamma }}_{i_1}\cdots {\mathrm{\Gamma }}_{i_m}\text{.}}& \text{(5.43)}\end{array}$$ In this normalization, $t=q^{k/2}$ (cf. the differential case). For instance, the so-called group case is for $k=1/2$ (in contrast to ${SL}_2$ considered above).

The Macdonald polynomials $p_{\lambda }$ for ${GL}_n$ satisfy the Macdonald eigenvalue problem: $$\begin{array}{cc}{M}_m{p}_{\lambda }=e_m(t^{n-1}{q}^{{\lambda }_1},\dots ,t^{-n+1}{q}^{{\lambda }_n})p_{\lambda }(m=0,1,\dots ,n),& \text{(5.44)}\end{array}$$ where $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ are partitions, i.e., sequences of integers ${\lambda }_i\in \mathbb{Z}$ such that ${\lambda }_1\ge {\lambda }_2\ge \dots {\lambda }_n\ge 0\text{.}$ Here $e_m$ elementary symmetric function of degree $m\text{.}$ Given $\lambda ,$ $p_{\lambda }=p_{\lambda }\left(x\right)$ is a symmetric polynomial in $x=(x_1,\dots ,x_n)$ of degree $\left|\lambda \right|=\sum _{i=1}^n{\lambda }_i$ in the form $$\begin{array}{cc}{p}_{\lambda }\left(x\right)=x_1^{{\lambda }_1}\cdots x_n^{{\lambda }_n}+\text{lower order terms.}& \text{(5.45)}\end{array}$$ The lower order terms are understood in the sense of the dominance ordering. Namely, a partition obtained from $\lambda$ by subtracting simple roots $(0,\dots ,0,1,-1,0,\dots ,0)$ is lower than $\lambda \text{.}$ For instance. $$\begin{array}{cc}({\lambda }_1,{\lambda }_2,\dots )>({\lambda }_1-1,{\lambda }_2+1,\dots )>({\lambda }_1-1,{\lambda }_2,{\lambda }_3+1\dots )>\cdots \text{.}& \text{(5.46)}\end{array}$$ We will use the abbreviation $$\begin{array}{cc}{t}^{2\rho }{q}^{\lambda }=(t^{n-1}{q}^{{\lambda }_1},\dots ,t^{-n+1}{q}^{{\lambda }_n}),& \text{(5.47)}\end{array}$$ where $2\rho =(n-1,n-3,\dots ,-n+1)\text{.}$ So $M_m{p}_{\lambda }=e_m\left(t^{2\rho }{q}^{\lambda }\right)p_{\lambda }\text{.}$ Using $k\text{:}$ $t=q^{k/2},$ and $t^{2\rho }{q}^{\lambda }=q^{k\rho +\lambda }\text{.}$

Given a partition $\lambda ,$ we set $$\begin{array}{cc}{\displaystyle {\pi }_{\lambda }\left(x\right)=\frac{p_{\lambda }\left(x\right)}{p_{\lambda }\left(t^{2\rho }\right)}=\frac{p_{\lambda }(x_1,\dots ,x_n)}{p_{\lambda }(t^{n-1},t^{n-3},\dots ,t^{-n+1})}\text{.}}& \text{(5.48)}\end{array}$$

Theorem 5.5 (Duality). For any partitions $\lambda$ and $\mu ,$ we have $$\begin{array}{cc}{\pi }_{\lambda }\left(t^{2\rho }{q}^{\mu }\right)={\pi }_{\mu }\left(t^{2\rho }{q}^{\lambda }\right)\text{.}& \text{(5.49)}\end{array}$$

This duality theorem implies the following Pieri formula.

Theorem 5.6. $$\begin{array}{cc}{\displaystyle e_m\left(x\right){\pi }_{\lambda }\left(x\right)=\sum _{\left|I\right|=m}\prod _{\underset{j\notin I}{i\in I}}\frac{t^{2(j-i)+1}{q}^{{\lambda }_i-{\lambda }_j}-t^{-1}}{t^{2(j-i)}{q}^{{\lambda }_i-{\lambda }_j}-1}{\pi }_{\lambda +e_I}\left(x\right),}& \text{(5.50)}\end{array}$$ where $e_I=\sum _{i\in I}{e}_i$ (sum of unit vectors).

Here the summation is taken only over subsets $I\subset \{1,2,\dots ,n\}$ $\text{(}\left|I\right|=m\text{)}$ such that $\lambda +e_I$ remain partitions (generally speaking, dominant). It happens automatically, since the coefficient of ${\pi }_{\lambda +e_I}\left(x\right)$ on the right vanishes unless $\lambda +e_I$ is dominant.

We can also determine the value of the Macdonald polynomial $p_{\lambda }\left(x\right)$ at $x=t^{2\rho }$ exactly by the method used in the $A_1$ case. The formula was conjectured by Macdonald and proved by Koornwinder. The above theorems (for ${GL}_n\text{)}$ are also due to Macdonald and Koornvvinder. See also [EKi1995]. For arbitrary roots they were established in my recent papers.

Our approach is based on the double Hecke algebras. The operators $M_m$ appear naturally using the operators ${\mathrm{\Delta }}_i$ from (4.32). The latter describe the action of the generators $Y_i$ in the induced representation ${\text{Ind}}_{H_Y}^{\mathscr{H}}(-)$ isomorphic to the algebra of Laurent polynomials $ℂ[X_1^{\pm 1},\dots ,X_n^{\pm 1}]\text{.}$ So the analogy with (5.26) is complete.

The double affine Hecke algebra (DAHA) $\mathscr{H}={\mathscr{H}}^{q,t}$ for ${GL}_n$ is the algebra generated by the following two commutative algebras of Laurent polynomials in $n$ variables: $$\begin{array}{cc}ℂ[X_1^{\pm 1},\dots ,X_n^{\pm 1}]\text{and}ℂ[Y_1^{\pm 1},\dots ,Y_n^{\pm 1}],& \text{(5.51)}\end{array}$$ and the Hecke algebra of type $A_{n-1}\text{:}$ $$\begin{array}{cc}\mathscr{H}=⟨T_1,\dots ,T_{n-1}⟩& \text{(5.52)}\end{array}$$ with the standard braid and quadratic relations. The remaining relations are as follows: $$\begin{array}{cc}{T}_i{X}_i{T}_i=X_{i+1}\hspace{0.17em}\text{(}i=1,\dots ,n-1\text{),}{T}_i{X}_j=X_j{T}_i\hspace{0.17em}\text{(}j\ne i,i+1\text{),}& \text{(5.53)}\\ T_i^{-1}{Y}_i{T}_i^{-1}=Y_{i+1}\hspace{0.17em}\text{(}i=1,\dots ,n-1\text{),}{T}_i{Y}_j=Y_j{T}_i\hspace{0.17em}\text{(}j\ne i,i+1\text{),}& \text{(5.54)}\\ Y_2^{-1}{X}_1{Y}_2{X}_1^{-1}=T_1^2,& \text{(5.55)}\\ \stackrel{\sim }{Y}{X}_j=q{X}_j\stackrel{\sim }{Y}\text{and}\stackrel{\sim }{X}{Y}_j=q^{-1}{Y}_j\stackrel{\sim }{X}\text{.}& \text{(5.56)}\end{array}$$ Here $\stackrel{\sim }{X}=\prod _{i=1}^n{X}_i$ and $\stackrel{\sim }{Y}=\prod _{i=1}^n{Y}_i\text{.}$ They commute with $\{T_1,\dots ,T_{n-1}\}$ thanks to 5.53 and 5.55.

When $q=1,$ $t=1$ we come to the elliptic or 2-extended Weyl group of type ${GL}_n$ due to Saito [Sai1985]. If $q=1$ and there are no quadratic relations, the corresponding group is the elliptic braid group $\text{(}{\pi }_1$ of the product of $n$ elliptic curves without the diagonals divided by ${𝕊}_n\text{).}$ It was calculated by Birman [Bir1969] and Scott.

Establishing the connection with (4.32), $X_i=e^{v_i},$ $T_i={\hat{T}}_i$ and $Y_i={\mathrm{\Delta }}_i$ give the so-called polynomial (or basic) representation of $\mathscr{H}\text{.}$

There is another version of this definition, using the element $\pi \text{.}$ It is introduced from the formula $$\begin{array}{cc}{Y}_1=T_1\cdots T_{n-1}{\pi }^{-1}& \text{(5.57)}\end{array}$$ and has the following commutation relations with $X_i$ and $T_i\text{:}$ $$\begin{array}{cc}\pi X_i=X_{i+1}\pi (i=1,\dots ,n-1),\pi X_n=q^{-1}{X}_1\pi & \text{(5.58)}\end{array}$$ and $$\begin{array}{cc}\pi T_i=T_{i+1}\pi (i=1,\dots ,n-2)\text{.}& \text{(5.59)}\end{array}$$ In the polynomial representation this element coincides with $\pi$ from Lemma 4.3. Note that it acts on the functions $X_i=e^{v_i}$ through the action of ${\pi }^{-1}$ on vectors $v\text{.}$ Considered formally, $\pi$ is the image of the element $P$ from Lemma 4.1 with respect to the Kazhdan-Lusztig automorphism, sending $T\to T^{-1},$ $Y\to Y^{-1},$ $t\to t^{-1}\text{.}$

Since $$\begin{array}{cc}{T}_{i-1}\cdots T_1{X}_1{T}_1\cdots T_{i-1}=X_i,T_1\cdots T_{i-1}{Y}_i{T}_{i-1}\cdots T_1=Y_1,& \text{(5.60)}\end{array}$$ we can reduce the list of generators. Namely, $$\begin{array}{cc}\mathscr{H}=⟨X_1,Y_1,T_1,\dots ,T_{n-1}⟩& \text{(5.61)}\end{array}$$ or $$\begin{array}{cc}\mathscr{H}=⟨X_1,\pi ,T_1,\dots ,T_{n-1}⟩\text{.}& \text{(5.62)}\end{array}$$

In terms of $\{T,\pi ,X\},$ the list of defining relations of $\mathscr{H}$ is as follows:

(a)	$X_i{X}_j=X_j{X}_i$ $\text{(}1\le i,j\le n\text{),}$
(b)	the braid relations and quadratic relations for $T_1,\dots ,T_{n-1},$
(c)	$\pi X_i=X_{i-1}\pi$ $\text{(}i=1,\dots ,n-1\text{)}$ and ${\pi }^n{X}_i=q^{-1}{X}_i{\pi }^n$ $\text{(}i=1,\dots ,n\text{),}$
(d)	$\pi T_i=T_{i+1}\pi$ $\text{(}i=1,\dots ,n-2\text{)}$ and ${\pi }^n{T}_i=T_i{\pi }^n$ $\text{(}i=1,\dots ,n-1\text{).}$

For instance, let us deduce (5.55) from these formulas. Substituting, the left hand side equals: $$\begin{array}{ccc}& & \left(T_1\pi T_{n-1}^{-1}\cdots T_2^{-1}\right)X_1\left(T_2\cdots T_{n-1}{\pi }^{-1}{T}_1^{-1}\right)X_1^{-1}\\ & =& T_1\pi T_{n-1}^{-1}\cdots T_2^{-1}\left(T_2\cdots T_{n-1}\right)X_1{\pi }^{-1}\left(T_1^{-1}{X}_1^{-1}{T}_1^{-1}\right)T_1\\ & =& T_1\left(\pi X_1{\pi }^{-1}{X}_2^{-1}\right)T_1=T_1^2\text{.}\end{array}$$ This representation, however, is not convenient from the viewpoint of the symmetry between $X_i$ and $Y_i,$ which will be discussed next. It is better to use $\left\{Y\right\}$ instead of $\pi \text{.}$

The algebra $\mathscr{H}$ contains the following two affine Hecke algebras: $${\mathscr{H}}_X^t=⟨X_1,\dots ,X_n,T_1,\dots ,T_{n-1}⟩\text{.}$$ $$\begin{array}{cc}{\mathscr{H}}_Y^t=⟨Y_1,\dots ,Y_n,T_1,\dots ,T_{n-1}⟩\text{.}& \text{(5.63)}\end{array}$$ They are isomorphic to each other by the correspondence $X_i\leftrightarrow Y_i^{-1}\text{.}$ This map can be extended to an anti-involution of $\mathscr{H}\text{.}$ It is a general statement which holds for any root systems.

Theorem 5.7. There exists an anti-involution $\varphi :{\mathscr{H}}^{q,t}\to {\mathscr{H}}^{q,t}$ such that $\varphi \left(X_i\right)=Y_i^{-1},$ $\varphi \left(Y_i\right)=X_i^{-1}$ for $i=1,\dots ,n,$ and $\varphi \left(T_i\right)=T_i$ $\text{(}i=1,\dots ,n-1\text{).}$ It preserves $q,t\text{.}$

		Proof.
	We need to check that the relation (5.55) is self-dual with respect to $\varphi \text{.}$ The other relations are obviously $\varphi \text{-invariant.}$ One has: $$\begin{array}{ccc}1& =& T_1^{-2}{Y}_2^{-1}{X}_1{Y}_2{X}_1^{-1}=T_1^{-1}\left\{Y_1^{-1}\left(T_1{X}_1{T}_1^{-1}\right)Y_1{T}_1^{-1}{X}_1^{-1}\right\}\\ & =& Y_1^{-1}{X}_2{T}_1^{-2}{Y}_1\left(T_1^{-1}{X}_1^{-1}{T}_1^{-1}\right)=Y_1^{-1}{X}_2{T}_1^{-2}{Y}_1{X}_2^{-1}\text{.}\end{array}$$ The latter can be rewritten as $Y_1{X}_2^{-1}{Y}_1^{-1}{X}_2=T_1^2,$ which is the $\varphi \text{-image}$ of (5.55). $\square$

Using this involution we can establish the duality theorem for the ${GL}_n$ case in the same way as we did in the $A_1$ case. Generalizing the theory to the case of arbitrary roots we can prove the Macdonald conjectures and much more. It gives a very convincing example of the power of the modern difference-operator methods.

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