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\title{Koornwinder-Macdonald polynomials}

\author{Jasper V. Stokman}

\address{Jasper V. Stokman,
KdV Institute for Mathematics, Universiteit van Amsterdam,
Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.}
\email{jstokman@wins.uva.nl}

\begin{document}


\maketitle

\section{Introduction}
The Macdonald \cite{Mac1} polynomials form a remarkable
family of multivariable orthogonal polynomials.
They have a wide range of applications in mathematics
and mathematical physics, which vary from representation
theory of quantum groups and Hecke algebras
to integrable systems and conformal field theory.

Important insight into the structure of Macdonald polynomials
was provided by Cherednik \cite{C1}--\cite{C3},
who realized the underlying symmetries
in terms of explicit difference-reflection operators.
His techniques have had (and still have) a tremendous impact on
the subject, and have led to proofs of several important conjectures regarding
the structure of Macdonald polynomials (the so-called Macdonald conjectures).

Koornwinder \cite{K}
added extra degrees of freedom into the theory, which has
led to a simultaneous generalization of all families of Macdonald polynomials
associated with classical root systems. These polynomials are nowadays
known as Koornwinder(-Macdonald) polynomials.

The Koornwinder
polynomials are multivariable analogues of the Askey-Wilson \cite{AW}
polynomials. The Askey-Wilson polynomials are the
most general family of
orthogonal polynomials which satisfy
a second order $q$-difference equation, see
e.g. \cite{GH} and references therein.
In the terminology of \cite{AA} and \cite{GH}, orthogonal
polynomials which satisfy a second order $q$-difference equation are
called classical (basic hypergeometric) orthogonal polynomials.


The central role of Askey-Wilson polynomials in the theory of
classical orthogonal polynomials seems to be
taken over by the Koornwinder polynomials in the multivariable
setting. First of all, the symmetric Koornwinder polynomials are
eigenfunctions of a large family of $q$-difference operators, which includes
an explicit second order $q$-difference operator,
see e.g. \cite{K} and \cite{vD1}.
Furthermore, the large scheme of families of classical orthogonal
polynomials which are obtainable as special cases or limit cases
of the Askey-Wilson polynomials (see \cite{KK}) generalizes
(at least partially) to the multivariable setting,
in which the Koornwinder polynomials are now on top of the scheme,
see e.g. \cite{KSt} and \cite{vDtrans}.


In my lecture I will discuss some of the main developments in the
theory of Macdonald polynomials. I will focus on the extension of
Cherednik's \cite{C1}--\cite{C3} and Macdonald's \cite{Mac1}
theory on Macdonald polynomials and affine Hecke algebras to the
Koornwinder set-up, which was developed in e.g.
\cite{N}, \cite{Sa1}, \cite{Sa2}, \cite{NS}, \cite{St} and \cite{NUKW}.
In particular,  I will discuss non-symmetric analogues
of the Koornwinder polynomials and their basic properties, such as
bi-orthogonality relations and duality.
I will keep track of the implications in
the one-variable setting, in which case we are dealing with
Askey-Wilson polynomials (see \cite{NS} and \cite{Sa2}).

In the next section I give a short
introduction to the subject by stating the main properties of the symmetric
Koornwinder polynomials ($q$-difference equations, duality,
orthogonality relations and quadratic norm evaluations),
without going into details of the proofs. I will keep track of
the one variable case, in which case the results reduce to well known
properties of the Askey-Wilson polynomials \cite{AW}. I will
furthermore try to indicate, without going into to many details,
how the transition from the symmetric
theory on Koornwinder polynomials to the non-symmetric theory arises
in a natural way. More details will be provided in the lecture itself.

{\it Acknowledgements:}
The author is supported by a fellowship from the Royal
Netherlands Academy of Arts and Sciences (KNAW).





\section{Koornwinder-Macdonald polynomials}
Let $n$ be a positive integer and let $V=\mathbb{R}^n$ be
Euclidean $n$-space with standard scalar product
$\bigl(.,.\bigr)$ and standard orthonormal basis
$\{\epsilon_i\}_{i=1}^n$ (where $\epsilon_i$ has zeros for all its coordinates,
except a one at the $i$th coordinate).
Let $\Lambda=\bigoplus_{i=1}^n\mathbb{Z}\epsilon_i$ be the lattice
of $V$ consisting of vectors with integer coordinates.


We assume for the moment that
$q,a,b,c,d,t$ are six indeterminates, and we let $\mathbb{K}$ be
the field of rational functions in $q,a,b,c,d,t$ over
$\mathbb{C}$. Later on we will fix the parameters $q,a,b,c,d,t$
to generic complex values and work over the ground field
$\mathbb{C}$.

Let $\mathcal{A}=\mathbb{K}[x_1^{\pm 1},\ldots, x_n^{\pm 1}]$ be the algebra
of Laurent polynomials in $n$ independent indeterminates $x_1,\ldots,x_n$.
A linear basis of $\mathcal{A}$ is given by the set of monomials
$\{x^{\lambda}\, | \, \lambda\in\Lambda\}$, where
\[ x^{\lambda}=x_1^{\lambda_1}x_2^{\lambda_2}\cdots
x_n^{\lambda_n},\qquad \lambda\in\Lambda.
\]

Let $S_n$ be the symmetric group in $n$ letters.
The finite Weyl group $W=S_n\ltimes (\pm 1)^n$ acts on $V$ by
permutations and sign changes of the coordinates. Since the lattice
$\Lambda\subset V$ is preserved under this action, we obtain an
induced action of $W$ on $\mathcal{A}$ by linear extension of
\[ w(x^{\lambda})=x^{w\lambda},\qquad \lambda\in\Lambda,\,\,\,
w\in W.
\]
In other words, $W$ acts on $\mathcal{A}$ be permutations and
inversions of the $n$ variables $x_1,\ldots,x_n$.
We denote $\mathcal{A}^W$
for the sub-algebra of $\mathcal{A}$ consisting of Laurent polynomials
$p\in\mathcal{A}$ satisfying $w\,p=p$ for all $w\in W$.

Let
$\Lambda^+\subset \Lambda$ be the subset consisting of partitions
of length $\leq n$:
\[\Lambda^+=\{\lambda=(\lambda_1,\ldots,\lambda_n)\in\Lambda \, |
\, \lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\geq 0\}.
\]
Then $\mathcal{A}^W$ has a linear basis
$\{ m_{\lambda} \, | \, \lambda\in\Lambda^+\}$,
with $m_{\lambda}$ the orbit sum defined by
\[m_{\lambda}(x)=\sum_{\mu\in W\lambda}x^{\mu},\qquad
\lambda\in\Lambda^+.
\]
Koornwinder's \cite{K} second order $q$-difference operator $L$ is now
defined by
\[ L=\sum_{j=1}^n\bigl(\phi_j(x)(\tau(\epsilon_j)-1)+\phi_j(x^{-1})
(\tau(-\epsilon_j)-1)\bigr),
\]
with the rational coefficient $\phi_j(x)$ given by
\[
\phi_j(x)=\frac{(1-ax_j)(1-bx_j)(1-cx_j)(1-dx_j)}
{(1-x_j^2)(1-qx_j^2)}
\prod_{i\not=j}\frac{(1-tx_ix_j)(1-tx_i^{-1}x_j)}
{(1-x_ix_j)(1-x_i^{-1}x_j)},
\]
and with $\tau(\mu)\in \hbox{End}_{\mathbb{K}}(\mathcal{A})$
($\mu\in\Lambda$) the $q$-difference operator defined by
\[\tau(\mu)\bigl(x^{\lambda}\bigr)=q^{(\mu,\lambda)}x^{\lambda},
\qquad \lambda,\mu\in\Lambda.
\]
Koornwinder \cite{K} showed that $L$ preserves $\mathcal{A}^W$
and proved that $L$ is triangular with respect to a particular partial
order on the orbit sums $\{m_{\mu} \, | \, \mu\in\Lambda^+\}$, with
diagonal terms given by
\[E_{\lambda}=\sum_{j=1}^n\bigl(q^{-1}abcdt^{2n-j-1}(q^{\lambda_j}-1)
+t^{j-1}(q^{-\lambda_j}-1)\bigr),\qquad \lambda\in\Lambda^+.
\]
We now set
\begin{equation}\label{xnul}
x_{0}=\bigl(a,at,\ldots,at^{n-1}\bigr),
\end{equation}
then the above arguments lead to the following theorem, see \cite{K}.
\begin{thm}\label{qdiff}
There exists a unique basis
$\{ P_{\lambda}  | \, \lambda\in\Lambda^+\}$
of $\mathcal{A}^W$, such that
\begin{enumerate}
\item[{\bf --}] $\bigl(LP_{\lambda}\bigr)(x)=
E_{\lambda}P_{\lambda}(x)$,
\item[{\bf --}] $P_{\lambda}(x_0)=1$
\end{enumerate}
for all $\lambda\in\Lambda^+$.
\end{thm}
The $W$-invariant Laurent polynomial $P_{\lambda}$
($\lambda\in\Lambda^+$) is called the {\it symmetric Koornwinder
polynomials of degree $\lambda$}. By the triangularity of $L$
it can be shown that the monomial $m_{\lambda}(x)$
is the highest order contribution of $P_{\lambda}(x)$
in its expansion of orbit sums $m_{\mu}(x)$ ($\mu\in\Lambda^+$)
with respect to a particular partial order on $\Lambda^+$,
which justifies calling $P_{\lambda}$ of degree
$\lambda$. The advantage of the chosen normalization for $P_{\lambda}$
(see the second property of $P_{\lambda}$ in theorem \ref{qdiff})
will become apparent when we
discuss the duality of the symmetric Koornwinder polynomials.

For $n=1$, the symmetric Koornwinder polynomial
$P_{\lambda}$ ($\lambda\in\mathbb{Z}_+$) is the
Askey-Wilson polynomial of degree $\lambda$:
\begin{equation}\label{onevariableAW}
P_{\lambda}(x)=
{}_4\phi_3\left(\begin{matrix}&q^{-\lambda},  q^{\lambda-1}abcd, ax, ax^{-1} \\
& ab, ac, ad \end{matrix}; q,q\right),\qquad
\lambda\in\mathbb{Z}_+,\,\,\, n=1
\end{equation}
(independent of $t$), where we used the standard basic
hypergeometric series notations as used in the book \cite{GR}
of Gasper and Rahman. Theorem \ref{qdiff} then reduces to
the well known fact that
the Askey-Wilson polynomials are eigenfunctions of the
Askey-Wilson second order $q$-difference operator, see \cite{AW}.

{}From now on we specialize the parameters $q,a,b,c,d,t$ to
generic complex values, and we
define dual parameters $\tilde{a}, \tilde{b}, \tilde{c}, \tilde{d}$ by
\begin{equation}
\tilde{a}=\sqrt{q^{-1}abcd}, \quad
\tilde{b}=ab/\tilde{a}, \quad \tilde{c}=ac/\tilde{a}, \quad
\tilde{d}=ad/\tilde{a},
\end{equation}
where $\sqrt{\cdot}$ is the branch of the square root which is
positive on $\mathbb{R}_{>0}$. We now observe that
the eigenvalue $E_{\lambda}$ of the symmetric Koornwinder
polynomial $P_{\lambda}$ is of the form
\begin{equation}\label{eigenvaluecomparison}
E_{\lambda}=\tilde{a}t^{n-1}\bigl(m_{\epsilon_1}(\gamma_{\lambda})-
m_{\epsilon_1}(\gamma_0)\bigr),\qquad \forall\,\lambda\in\Lambda^+,
\end{equation}
where
\begin{equation}\label{gamma}
\gamma_{\lambda}=\bigl(\tilde{a}q^{\lambda_1},
\tilde{a}tq^{\lambda_2},\ldots
\tilde{a}t^{n-1}q^{\lambda_n}\bigr), \qquad
\lambda\in\Lambda^+.
\end{equation}
The expression \eqref{eigenvaluecomparison} for the eigenvalues of $L$
already reveals the
underlying theory of non-symmetric analogues of the Koornwinder
polynomials. To be a bit more precise, it turns out that
the second order $q$-difference operator $L$ can be decomposed
in a similar manner as the
decomposition \eqref{eigenvaluecomparison}
of the eigenvalue $E_{\lambda}$. Indeed,
Noumi \cite{N} constructed $n$ pair-wise
commuting, invertible linear operators $Y_1,\ldots,Y_n\in
\hbox{End}_{\mathbb{C}}(\mathcal{A})$ so that
\begin{equation}\label{LY}
L=\tilde{a}t^{n-1}
\bigl(m_{\epsilon_1}(Y)-m_{\epsilon_1}(\gamma_0)\bigr)|_{\mathcal{A}^W},
\end{equation}
where we have written $p(Y)=\sum_{\lambda\in\Lambda}c_{\lambda}Y^{\lambda}$
($Y^{\lambda}=Y_1^{\lambda_1}Y_2^{\lambda_2}\cdots Y_n^{\lambda_n}$)
for $p(x)=\sum_{\lambda\in\Lambda}c_{\lambda}x^{\lambda}\in\mathcal{A}$.
The $Y$-operators $Y_i$ can be explicitly expressed in terms of
difference-reflection operators. Noumi \cite{N} also showed that
the $q$-difference operator $p(Y)|_{\mathcal{A}^W}$
preserves the sub-algebra $\mathcal{A}^W$ of $W$-invariant Laurent
polynomials when $p\in\mathcal{A}^W$.
These observations immediately lead
to a large family of $q$-difference operators diagonalizing the symmetric
Koornwinder polynomials. Indeed, let $p\in \mathcal{A}^W$ and
$\lambda\in\Lambda^+$, then $p(Y)P_{\lambda}\in
\mathcal{A}^W$, and by \eqref{LY} and theorem \ref{qdiff},
\[L\bigl(p(Y)P_{\lambda}\bigr)=
p(Y)\bigl(LP_{\lambda}\bigr)=
\bigl(m_{\epsilon_1}(\gamma_{\lambda})-m_{\epsilon_1}(\gamma_0)\bigr)
p(Y)P_{\lambda},
\]
hence $p(Y)P_{\lambda}$ is a constant multiple of
$P_{\lambda}$ (in fact, the eigenvalue turns out to be
$p(\gamma_{\lambda})$). In other words,
all the $q$-difference operators $p(Y)|_{\mathcal{A}^W}$
($p\in \mathcal{A}^W$) are diagonalized by the symmetric Koornwinder
polynomials, and
the points $\gamma_{\lambda}$ ($\lambda\in\Lambda^+$)
form the spectrum of the
commutative sub-algebra $\{p(Y)|_{\mathcal{A}^W} \, | \, p\in\mathcal{A}^W\}$
of $\hbox{End}_{\mathbb{C}}(\mathcal{A}^W)$.
The transition to the non-symmetric theory is now
completely natural, since it relates to the problem of simultaneously
diagonalizing the $Y$-operators $Y_i$ ($i=1,\ldots,n$), regarded now
as pair-wise commuting linear operators on $\mathcal{A}$.

If we replace the
role of the parameters $(a,b,c,d)$ by dual parameters
$(\tilde{a},\tilde{b},\tilde{c},\tilde{d})$,
the spectral point $\gamma_{\lambda}$
becomes
\[
x_{\lambda}=\bigl(aq^{\lambda_1}, atq^{\lambda_2},\ldots,
at^{n-1}q^{\lambda_n}\bigr),\qquad \lambda\in\Lambda^+
\]
(observe that this is compatible with the definition of $x_0$
in \eqref{xnul}). Let $\widetilde{P}_{\lambda}$ be the
symmetric Koornwinder polynomial
of degree $\lambda$ in
which the parameters $(a,b,c,d)$ are replaced by
$(\tilde{a},\tilde{b},\tilde{c},\tilde{d})$. The duality of the symmetric
Koornwinder polynomial $P_{\lambda}(x)$
between its geometric parameter $x$ and its spectral parameter $\lambda$
can then be formulated
as follows.

\begin{thm}\label{dualitysymm}
The symmetric Koornwinder polynomials satisfy
\[ P_{\lambda}(x_{\mu})=\widetilde{P}_{\mu}(\gamma_{\lambda}),
\qquad \forall\lambda,\mu\in\Lambda^+.
\]
\end{thm}
This duality theorem was proved by van Diejen \cite{vD1}
for a sub-family of the Koornwinder polynomials. The proof
for arbitrary parameter values was given by Sahi \cite{Sa1}. In
Sahi's \cite{Sa1} proof the non-symmetric theory and the associated
algebra of symmetries (the so-called double affine Hecke algebra)
play an important role. In fact, duality is closely related to the
existence of an isomorphism of the
double affine Hecke algebra which interchanges
``multiplication by $x_i$'' with ``acting by $Y_i$'' for all
$i=1,\ldots,n$.

For $n=1$, theorem \ref{dualitysymm} is a direct
consequence of the explicit expression \eqref{onevariableAW}
for the Askey-Wilson polynomials.

Combining theorem \ref{qdiff}, theorem
\ref{dualitysymm} and \eqref{eigenvaluecomparison}
leads to the three term recurrence
relation for the dual symmetric Koornwinder polynomials
$\widetilde{P}_{\lambda}$ ($\lambda\in\Lambda^+$). It
is given by the following identity in
$\mathcal{A}^W$,
\begin{equation*}
\begin{split}
\sum_{j=1}^n\left(\phi_j(x_{\lambda})
\bigl(\widetilde{P}_{\lambda+\epsilon_j}(x)-
\widetilde{P}_{\lambda}(x)\bigr)
+\right.&\left.
\phi_j(x_{\lambda}^{-1})\bigl(\widetilde{P}_{\lambda-\epsilon_j}(x)-
\widetilde{P}_{\lambda}(x)\bigr)\right)=\\
&=\tilde{a}t^{n-1}\bigl(m_{\epsilon_1}(x)-
m_{\epsilon_1}(\gamma_0)\bigr)\widetilde{P}_{\lambda}(x)
\end{split}
\end{equation*}
for $\lambda\in\Lambda^+$, where the contribution of the
term
$\phi_j(x_{\lambda}^{\pm 1})
\bigl(\widetilde{P}_{\lambda\pm\epsilon_j}(x)-
\widetilde{P}_{\lambda}(x)\bigr)$ in the left hand side
is taken to be zero if $\lambda\pm\epsilon_j\not\in\Lambda^+$.
For $n=1$, this gives the three term recurrence relation for the
Askey-Wilson polynomials, see \cite{AW}.

In order to state the orthogonality relations for the symmetric
Koornwinder polynomials, we have to assume that $|q|<1$.
We furthermore assume (for convenience) that $a,b,c,d$ and $t$
have moduli less than one.
We define a weight function $\Delta(x)$ by
\begin{equation*}
\begin{split}
\Delta(x)=&\prod_{i=1}^n\frac{\bigl(x_i^2,x_i^{-2};q\bigr)_{\infty}}
{\bigl(ax_i,ax_i^{-1}, bx_i, bx_i^{-1}, cx_i, cx_i^{-1}, dx_i,
dx_i^{-1};q\bigr)_{\infty}}\\
&\qquad\times\prod_{1\leq i<j\leq n}
\frac{\bigl(x_ix_j, x_ix_j^{-1}, x_i^{-1}x_i,
x_i^{-1}x_j^{-1};q\bigr)_{\infty}}
{\bigl(tx_ix_j, tx_ix_j^{-1}, tx_i^{-1}x_j, tx_i^{-1}x_j^{-1}
;q\bigr)_{\infty}},
\end{split}
\end{equation*}
where we used the standard notations for (products of) $q$-shifted
factorials, see \cite{GR}. Observe that $\Delta(x)$ is $W$-invariant,
where the finite Weyl group $W$ acts by permutations and
inversions of the variables $x_i$ ($i=1,\ldots,n$).
Furthermore, the weight function $\Delta(x)$ is positive for $x\in
\mathbb{T}^n$ when $q,a,b,c,d$ and $t$ are
also assumed to be positive real, where $\mathbb{T}$ is the
unit circle in the complex plane.
We define now a (non-degenerate)
bilinear form $\langle .,. \rangle$ on $\mathcal{A}^W$ by
\begin{equation}
\langle p,p^\prime\rangle=\frac{1}{(2\pi
i)^n}\iint_{x\in\mathbb{T}^n}p(x)p^\prime(x)\Delta(x)\frac{dx}{x},\qquad
p, p^\prime\in\mathcal{A}^W,
\end{equation}
where $\frac{dx}{x}=\frac{dx_1}{x_1}\cdots \frac{dx_n}{x_n}$
and $\mathbb{T}$ is positively oriented.
Then the second order $q$-difference operator $L$ is formally
symmetric with respect to $\langle .,. \rangle$. This leads
to Koornwinder's \cite{K} orthogonality relations for the
polynomials $P_{\lambda}$ ($\lambda\in\Lambda^+$):
\begin{thm}\label{ortho}
We have
\[
\langle P_{\lambda},P_{\mu}\rangle=0,\qquad
\lambda,\mu\in\Lambda^+,\,\,\, \lambda\not=\mu.
\]
\end{thm}
For $n=1$, the orthogonality relations of theorem \ref{ortho}
reduce to the orthogonality relations
for the Askey-Wilson polynomials, see \cite{AW}.

We define $w(x_{\lambda}^{-1})$ for $\lambda\in\Lambda^+$
by
\begin{equation}
w(x_{\lambda}^{-1})=\underset{x_1=x_{\lambda,1}^{-1}}
{\hbox{Res}}\left(\underset{x_2=x_{\lambda,2}^{-1}}{\hbox{Res}}
\left(\cdots \underset{x_n=x_{\lambda,n}^{-1}}{\hbox{Res}}
\left(\frac{\Delta(x)}{x_1\cdots x_n}\right)\cdots \right)\right),
\end{equation}
where $x_{\lambda,i}$ is the $i$th coordinate of $x_{\lambda}$.
We write $\widetilde{w}(\gamma_{\lambda}^{-1})$
($\lambda\in\Lambda^+$) for the discrete weight $w(x_{\lambda}^{-1})$
in which the parameters $(a,b,c,d)$ are replaced by the
corresponding dual parameters $(\tilde{a},\tilde{b},\tilde{c},\tilde{d})$.
The quadratic norms $\langle P_{\lambda}, P_{\lambda}\rangle$
($\lambda\in\Lambda^+$) for the
symmetric Koornwinder polynomials can now be evaluated as follows.
\begin{thm}\label{norms}
We have
\[ \frac{\langle P_{\lambda},
P_{\lambda}\rangle}{\langle 1,1\rangle}=
\frac{\widetilde{w}(\gamma_0^{-1})}{\widetilde{w}(\gamma_{\lambda}^{-1})},
\qquad \lambda\in\Lambda^+.
\]
\end{thm}
For $n=1$, this theorem reduces to the
quadratic norm evaluations for the Askey-Wilson polynomials, up to
the evaluation of the Askey-Wilson integral $\langle
1,1\rangle$ (see \cite{AW} and \cite{NS}).
The Askey-Wilson integral was evaluated
by Askey and Wilson \cite{AW}, while its
multivariable analogue $\langle 1,1\rangle$
was evaluated by Gustafson \cite{G}.

The quadratic norms in theorem \ref{norms} were derived for a
sub-family of the symmetric Koornwinder polynomials by van Diejen
\cite{vD1}, and subsequently extended to arbitrary parameter values by Sahi
\cite{Sa1}. The description of the quadratic norms
in terms of multiple residues of the weight function was given in \cite{St}.
In \cite{St} theorem \ref{norms} is derived as a direct consequence
of the bi-orthogonality relations and the corresponding diagonal
term evaluations for the non-symmetric analogues of the
Koornwinder polynomials. Proofs in the non-symmetric set-up are usually
easier, mainly due to the fact that the algebraic properties
of the associated algebra of symmetries (the so-called
double affine Hecke algebra) are easier to understand. In my lecture I plan to
discuss the non-symmetric theory for the Koornwinder set-up
in more detail, and I will show
how it relates to the results on symmetric Koornwinder polynomials
as discussed in this section. The lecture will be mainly based on the
papers \cite{N}, \cite{Sa1}, \cite{Sa2}, \cite{NS} and \cite{St}.


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\end{thebibliography}

\end{document}


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