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\begin{document}
\author{Guy Roos}
\address{191024 S. Petersburg, Nevski pr. 113/4-53 Russia}
\email{roos@mx.amss.ac.cn}
\title{Generalization of an integral of Hua Loo-keng}
\date{February 2003}
\subjclass{32H10}
\keywords{Cartan domain, Bergman kernel, holomorphic automorphism group, bounded
symmetric domain}
\maketitle


\thispagestyle{empty}
In his book ``Harmonic Analysis in Classical Domains'', Hua Loo-keng has
computed integrals of the following type%
\[
\int_{\frak{{R}}_{I}}\det(I-Z\overline{Z}^{\prime})^{\lambda\ }\overset{.}{Z},
\]
where
\[
\frak{R}_{I}=\left\{  Z\in\mathcal{M}_{m,n}(\mathbb{{C})}|I_{m}-Z^{t}%
\overline{Z}>>0\right\}
\]
is the ``generalized unit ball'' of $m\times n$ complex matrices. Analogous
integrals have been computed by Hua for all classical bounded symmetric
domains. For classical real simple Lie groups, similar integrals have been
computed recently by Yu. Neretin.

If $\Omega$ is any bounded irreducible complex symmetric domain (including the
two exceptional cases), the true generalization of these integrals appears to
be
\[
J(s)=\int_{\Omega}N(x,x)^{s}\alpha^{n},
\]
where $N(x,x)$ denotes the \emph{generic norm} of $\Omega$ and $\alpha$ is a
suitable volume form . We will call $J(s)$ the \emph{Hua integral}. Using an
integral of Selberg and the presentation of bounded symmetric domains
\emph{via} Jordan triple systems, we show that $J(s)$ is, as a function of
$s$, the inverse of polynomial $\chi(s)$, with integer or half-integer
coefficients, whose degree is the complex dimension of $\Omega$.

The main points of the theory of Jordan triple systems will be recalled.

We are asking about an interpretation of the polynomial $\chi$, which is
related with restricted root systems of type $BC_{r}$, and seems to carry all
the information about the corresponding symmetric domain.
\end{document}