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%% Translation via Omnimark script a2l, May 26, 1998 (all in one day!)
\controldates{9-JUN-1998,9-JUN-1998,9-JUN-1998,9-JUN-1998}
 
\documentclass{era-l}


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\theoremstyle{remark}
\newtheorem*{remark1}{Remarks}


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\newcommand{\ph}{\varphi }
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\newcommand{\Hom}{\operatorname {Hom}}
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\newcommand{\Un}{U_q \frak s\frak l _n }


\begin{document}

\title{On Cherednik--Macdonald--Mehta identities}
\author{Pavel Etingof}
\address{Department of Mathematics, Harvard University, Cambridge, MA 02138}
\email{etingof@math.harvard.edu}

\author{Alexander Kirillov, Jr.}
\address{Department of Mathematics, MIT, Cambridge, MA 02139}
\email{kirillov@math.mit.edu}
\issueinfo{4}{07}{}{1998}
\dateposted{June 11, 1998}
\pagespan{43}{47}
\PII{S 1079-6762(98)00045-6}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}
\commby{David Kazhdan}
\date{April 14, 1998}
\subjclass{Primary 05E35}
\keywords{Macdonald polynomials}
\begin{abstract}In this note we give a proof of Cherednik's generalization of
  Mac\-donald--Mehta identities for the root system $A_{n-1}$, using
  representation theory of quantum groups. These identities
  give an explicit formula for the integral of a product of Macdonald
  polynomials with respect to a ``difference analogue of the Gaussian
  measure''.  They were suggested by Cherednik, who also gave a proof
  based on representation theory of affine Hecke algberas; our
  proof gives a nice interpretation for these  identities in terms of
  representations of quantum groups and  seems to be simpler than
  that of Cherednik.
\end{abstract}
\maketitle

\section*{Introduction}

In this note we give a proof of Cherednik's generalization of
Macdonald--Mehta identities for the root system $A_{n-1}$, using
representation theory of quantum groups. These identities, suggested
and proved in \cite{Ch2}, give an explicit formula for the integral of
a product of Macdonald polynomials with respect to a ``difference
analogue of the Gaussian measure''. They can be written for any
reduced root system, or, equivalently, for any semisimple complex Lie
algebra $\fg $. Assuming for simplicity that $\fg $ is simple and
simply laced, these identities are given by the following formula:
\begin{equation}\label{eq:1}
\begin{split}
\frac{1}{|W|}\int \de _{k} \overline{\de _{k}} P_{\gl }\overline{P_{\mu }}\gamma \, dx
&=q^{\gl ^{2} +(\mu , \mu +2k\rho )} P_{\mu }(q^{-2(\gl +k\rho )})\\
&\quad \times q^{-2k(k-1)|R_{+}|}\prod _{\al \in R_{+}}\prod _{i=0}^{k-1}
				(1-q^{2(\al , \gl +k\rho )+2i})
\end{split} 
\end{equation}
where $\gl , \mu $ are dominant integral weights, 
$k$ is a positive integer, $P_{\gl }$ are Macdonald
polynomials associated with the corresponding root system, with
parameters $q^{2}, t=q^{2k}$(see
\cite{M1}, \cite{M2} or a review in \cite{Kir2}), $\de _{k}$ is the $q$-analogue
of $k$th power of the Weyl denominator $\de =\de _{1}$:
\begin{equation}\label{eq:2}
\de _{k}=\prod _{\al \in R^{+}}\prod _{i=0}^{k-1}(e^{\al /2}-q^{-2i}e^{-\al /2}),
\end{equation}
and $\gamma $ is the Gaussian, which we define by
\begin{equation}\label{eq:3}
\gamma =\sum _{\gl \in P} e^{\gl }q^{\gl ^{2}},
\end{equation}
where $P$ is the weight lattice. We consider
$\gamma $ as a formal series in $q$ with coefficients from the group
algebra of the weight lattice. In a more standard terminology $\gamma $
is called the theta-function of the lattice $P$. All other notations,
which are more or less standard, will be explained below.

These identities were  formulated in the form we use in a
paper of Cherednik \cite{Ch2}, who also proved them using double
affine Hecke algebras (note: our notations are somewhat different from
Cherednik's ones). We refer the reader to \cite{Ch2} for the
discussion of the history of these identities and their role in
difference harmonic analysis. 

The proof uses  the following
identity for the Gaussian, due to Kostant \cite{Kos}:
\begin{equation}\label{eq:4}
\gamma =\biggl (\prod _{\al \in R_{+}}(1-q^{2(\al , \rho )})
	\biggr )
	\sum _{\nu \in P_{+}}q^{(\nu , \nu +2\rho )}(\dimq L_{\nu }) \chi _{\nu }.
\end{equation}
Here $\chi _{\nu }$ is the character of the  irreducible
finite-dimensional module $L_{\nu }$ over $\fg $, 
and $\dimq L_{\nu }:=\chi _{\nu }(q^{2\rho })$ is the quantum dimension 
of $L_{\nu }$. 

\subsection*{Notations }We use the same notations as in \cite{EK1}, \cite{EK2} with the following
exceptions: we replace $q$ by $q^{-1}$ (note that this does not change
Macdonald's polynomials) and we use the notation $\ph _{\gl }$ for
``generalized characters'' (see below), reserving the notation
$\chi _{\gl }$ for usual (Weyl) characters. In particular, we define
$\overline{e^{\gl }}=e^{-\gl }, \bar q=q$, and for $f\in \Cq [P]$, we define
$f(q^{\gl }), \gl \in P$ by $e^{\mu }(q^{\gl })=q^{(\mu , \gl )}$. For brevity, we
also write $\gl ^{2}$ for $(\gl , \gl )$. Finally, we denote by $\int \
dx:\Cq [P]\to \Cq $ the functional of taking the constant term: $\int e^{\gl }\, dx=\de _{\gl , 0}$.

\section*{ The proof }
In this section, we give a proof of the Cherednik--Macdonald--Mehta
identities \eqref{eq:1} for $\fg =\sln $. The proof is based on the realization of
Macdonald's polynomials as ``vector-valued characters'' for the
quantum group $\Un $, which was given in \cite{EK1}. For the sake of
completeness, we briefly outline these results here, referring the
reader to the original paper for a detailed exposition.

Let us fix $k\in \bZ _{+}$ and denote by $U$ the finite-dimensional
representation of $\Un $ with highest weight $n(k-1)\omega _{1}$, where
$\omega _{1}$ is the first fundamental weight. We identify the zero
weight subspace $U[0]$, which is one-dimensional, with $\Cq $. 

For $\gl \in P_{+}$, we denote by $\Phi _{\gl }$ the unique intertwiner 
\begin{equation*}\Phi _{\gl }:L_{\gl +(k-1)\rho }\to L_{\gl +(k-1)\rho }\otimes U
\end{equation*}
and define the generalized character $\ph _{\gl }\in \Cq [P]\otimes U[0]\simeq \Cq [P]$  by 
$\ph _{\gl }(q^{x})=\Tr _{L_{\gl +(k-1)\rho }}(\Phi _{\gl }q^{x})$.

We can now summarize the results of \cite{EK1} as follows:
\begin{equation}\label{eq:5}
\begin{gathered}
\ph _{0}=\prod _{\al \in R_{+}}\prod _{i=1}^{k-1}(e^{\al /2}-q^{-2i}e^{-\al 
/2})=\de _{k}/\de, \\
\ph _{\gl }/\ph _{0}=P_{\gl },\end{gathered}
\end{equation}
where $P_{\gl }$  is the Macdonald polynomial with parameters
$q^{2}, t=q^{2k}$. 

We can also rewrite Macdonald's inner product in terms of the
generalized characters as follows. Recall that  Macdonald's  inner product 
on $\Cq [P]$ is defined by 
\begin{equation*}\langle f, g\rangle _{k}=\frac{1}{|W|}\int \de _{k}\overline{\de _{k}}f\bar g \, dx
\end{equation*}
(this differs by a certain power of $q$ from the original definition
of Macdonald).  Obviously, one has
\begin{equation*}\langle P_{\gl }, P_{\mu }\rangle _{k}=\langle \ph _{\gl }, \ph _{\mu }\rangle _{1}.
\end{equation*}
In order to rewrite this in terms of representation theory, let
$\omega $ be the Cartan involution in $\Un $ (see \cite{EK1}). For a
$\Un $-module $V$, we denote by $V^{\omega }$ the same vector space but
with the action of $\Un $ twisted by $\omega $. Note that for
finite-dimensional $V$, we have $V^{\omega }\simeq V^{*}$ (not
canonically). Similarly, for an intertwiner $\Phi :L\to L\otimes U$ we
denote by $\Phi ^{\omega }$ the corresponding intertwiner $L^{\omega }\to U^{\omega }\otimes L^{\omega }$. Finally, for $\Phi _{1}:L_{1}\to L_{1}\otimes U,
\Phi _{2}:L_{2}\to L_{2}\otimes U$, define $\Phi _{1}\odot \Phi _{2}^{\omega }\in \End (L_{1}\otimes L_{2}^{\omega })$
as the composition $L_{1}\otimes L_{2}^{\omega }\to L_{1}\otimes U\otimes U^{\omega }\otimes L_{2}^{\omega }\to L_{1}\otimes L_{2}^{\omega }$, where the first arrow is given by $\Phi _{\gl }\otimes \Phi _{\mu }^{\omega }$, and the second by the invariant
pairing $U\otimes U^{\omega }\to \Cq $
(which is unique up to a constant).  Then it was shown in \cite{EK1} that 
\begin{equation*}(\ph _{\gl }\overline{\ph _{\mu }})(q^{x})=\Tr _{V}((\Phi _{\gl }\odot \Phi ^{\omega }_{\mu })
q^{\Delta (x)})
=\sum _{\nu \in P_{+}}\chi _{\nu }(q^{x})C_{\gl \mu }^{\nu }\end{equation*}
where $V=L_{\gl +(k-1)\rho }\otimes L^{\omega }_{\mu +(k-1)\rho }$ and
$C_{\gl \mu }^{\nu }$ is the trace of $\Phi _{\gl }\odot \Phi _{\mu }$ acting in the
multiplicity space $\Hom (L_{\nu }, V)$. 
As a corollary, we get the following result:
\begin{equation}\label{eq:6}
\frac{1}{|W|}\int \de \bar \de \ph _{\gl }\overline{\ph _{\mu }}
\bigl (\sum _{\nu \in P^{+}}a_{\nu }\chi _{\nu }\bigr ) \, dx
= \sum _{\nu \in P^{+}} a_{\nu ^{*}} C_{\gl \mu }^{\nu },
\end{equation}
where $\nu ^{*}=-w_{0}(\nu )$ is the highest weight of the module $(L_{\nu })^{*}$
(here $w_{0}$ is the longest element of the Weyl
group).

Of course, the coefficients $C_{\gl \mu }^{\nu }$ are very difficult to
calculate. However, the formula above is still useful. For example, it
immediately shows that $\langle \ph _{\gl }, \ph _{\mu }\rangle _{1}=0$ unless $\gl =\mu $,
which was the major part of the proof of the formula
$\ph _{\gl }/\ph _{0}=P_{\gl }$ in \cite{EK1}. It turns out that this formula also
allows us to prove the Cherednik--Macdonald--Mehta identities.

\begin{thm}\label{theorem1} Let $\ph _{\gl }$ be the renormalized Macdonald
polynomials for the root system $A_{n-1}$ given by \eqref{eq:5}, and let
$\gamma $ be the Gaussian \eqref{eq:3}. Then
\begin{equation}\label{eq:7}
\begin{split}
\frac{1}{|W|} \int \de \overline{\de } \ph _{\gl }\overline{\ph _{\mu }} \gamma \, dx
&= q^{(\gl +k\rho )^{2}}q^{(\mu +k\rho )^{2}}\ph _{\mu }(q^{-2(\gl +k\rho )})\\
&\quad \times \biggl (\prod _{\al \in R_{+}}(1-q^{2(\al , \rho )})
	\biggr )
q^{-2\rho ^{2}}\|P_{\gl }\|^{2} \dimq L_{\gl +(k-1)\rho },
\end{split} 
\end{equation}
where $\|P_{\gl }\|^{2}=\langle P_{\gl }, P_{\gl }\rangle _{k}$. 
\end{thm}


\begin{proof} From \eqref{eq:6} and \eqref{eq:4}, we get 
\begin{equation}\label{eq:8}
\int \de \overline{\de } \ph _{\gl }\overline{\ph _{\mu }} \gamma \, dx= 
\biggl (\prod _{\al \in R_{+}}(1-q^{2(\al , \rho )})
	\biggr )\sum _{\nu \in P^{+}} q^{(\nu , \nu +2\rho )}(\dimq L_{\nu })
	C_{\gl \mu }^{\nu }.
\end{equation}

On the other hand, let $C$ be the Casimir element for $\Uq $ discussed
above.   Consider the intertwiner $(\Phi _{\gl }\odot \Phi _{\mu }^{\omega })
\Delta (C):V\to V$, where, as before, $V=L_{\gl +(k-1)\rho }\otimes L^{\omega }_{\mu +(k-1)\rho }$. Then it follows from $C|_{L_{\gl }}=q^{(\gl ,
\gl +2\rho )}$ that
\begin{equation*}\Tr _{V}((\Phi _{\gl }\odot \Phi _{\mu }^{\omega })
\Delta (C) \Delta (q^{2\rho }))=\sum _{\nu \in P_{+}}C_{\gl \mu }^{\nu }q^{(\nu ,
\nu +2\rho )}\dimq L_{\nu },\end{equation*}
which is exactly the sum on the right-hand side of \eqref{eq:8}. On the other
hand, using 
$\Delta (C)=(C\otimes C)(R^{21}R)$, we can write 
\begin{equation*}\begin{split}
&\Tr _{V}((\Phi _{\gl }\odot \Phi _{\mu }^{\omega })
\Delta (C) \Delta (q^{2\rho }))\\
&\qquad=q^{-2\rho ^{2}}q^{(\gl +k\rho )^{2}}q^{(\mu +k\rho )^{2}}
\Tr _{V}((\Phi _{\gl }\odot \Phi _{\mu }^{\omega }) 
(R^{21}R) \Delta (q^{2\rho })).\end{split}
\end{equation*}
This last trace can be calculated, which was done in \cite[Corollary~4.2]{EK2}, and the answer is given by 
\begin{equation*}\Tr _{V}((\Phi _{\gl }\odot \Phi _{\mu }^{\omega }) (R^{21}R) \Delta (q^{2\rho }))
=\ph _{\mu }(q^{-2(\gl +k\rho )})\|P_{\gl }\|^{2}\dimq L_{\gl +(k-1)\rho }.
\end{equation*}
Combining these results, we get the statement of the theorem.
\end{proof}


The norms $\|P_{\gl }\|^{2}$ appearing on the right-hand side of \eqref{eq:7}
are
given by Macdonald's inner
product identities 
\begin{equation*}\|P_{\gl }\|^{2}=\prod _{\al \in R_{+}}\prod _{i=1}^{k-1}
\frac{1-q^{-2 (\al , \gl +k\rho )-2i}}{1-q^{-2 (\al , \gl +k\rho )+2i}},
\end{equation*}
which were conjectured in  \cite{M1}, \cite{M2} and proved for the root system
$A_{n-1}$ 
by Macdonald himself \cite{M3}; see also \cite{EK2} for the proof based
on representation theory of $\Un $, and \cite{Ch1} or a review in
\cite{Kir2} for a proof for arbitrary root systems.  Using this formula
and rewriting the statement of Theorem~\ref{theorem1} in terms of Macdonald
polynomials $P_{\gl }$ rather than $\ph _{\gl }$, we get the
Cherednik--Macdonald--Mehta identities \eqref{eq:1}.

\begin{remark1}
1. Note that the left-hand side of \eqref{eq:7} is symmetric in $\gl ,
\mu $. Thus, the same is true for the right-hand side, which is exactly the
statement of Macdonald's symmetry identity (compare with the proof in
\cite{EK2}).

2. The proof of Cherednik--Macdonald--Mehta identities given above
easily generalizes to the case when $q$ is a root of unity (see
\cite{Kir1} for the discussion of the appropriate
representation-theoretic setup). In this case, we need to replace the
set $P_{+}$ of all integral dominant weights by an appropriate (finite)
Weyl alcove $C$ (see \cite{Kir1}), and the integral $\int \de \bar \de f\, dx$
should be replaced by 
$\text{const}\sum _{\gl \in C}f(q^{2(\gl +\rho )})\dimq L_{\gl }$.
Using the following obvious property of the Gaussian:
\begin{equation*}\gamma (q^{2(\gl +\rho )})=q^{-(\gl , \gl +2\rho )}\gamma (q^{2\rho })
\end{equation*}
(which in this case coincides with formula (1.7)  in
\cite{Kir1}), 
it is easy to see that in this case the
Cherednik--Macdonald--Mehta identities are equivalent to
\begin{equation*}S^{-1}T^{-1}S=TST,
\end{equation*}
where the matrices $S, T$  are defined in \cite[Theorem~5.4]{Kir1}. This
identity is part of a more general result, namely, that these
matrices $S, T$ give a projective representation of the modular group
$SL_{2}(\bZ )$ on the space of generalized characters 
(see \cite[Theorem 1.10]{Kir1} and references therein). 
\end{remark1}


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\end{document}