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Hua-type integrals over unitary groups and over projective limits of unitary groups. (English) Zbl 1019.43008

The author studies natural maps from $U\left(n\right)$ to $U\left(n-1\right)$ for which the image of the normalized Haar measure of $U\left(n\right)$ is that of $U\left(n-1\right)$. This makes it possible to define probability measures on infinite dimensional unitary groups. A family of integrals relative to these measures are evaluated. Let us state more precisely some of the main results of the paper. For $K=R$, $C$, or $H$, ${U}^{0}\left(n,K\right)$ is the group $SO\left(n\right)$, $U\left(n\right)$ or $Sp\left(n\right)$. The author considers the map

${\gamma }^{m}:{U}^{0}\left(n,K\right)\to {U}^{0}\left(n-m,K\right),$
$g=\left(\begin{array}{cc}P& Q\\ R& T\end{array}\right)↦T-R{\left(1+P\right)}^{-1}Q,$

for an $\left(m+\left(n-m\right)\right)×\left(m+\left(n-m\right)\right)$ block representation of $g$. It is proved that the image under the map ${\gamma }^{m}$ of the normalized Haar measure ${\sigma }_{n}$ of ${U}^{0}\left(n,K\right)$ is the Haar measure ${\sigma }_{n-m}$. Further the author considers the map

${\xi }_{m}:{U}^{0}\left(n,K\right)\to {U}^{0}\left(n-m,K\right)×{B}_{m},$
$g↦\left({\gamma }^{m}\left(g\right),{\left[g\right]}_{m}\right),$

where ${\left[g\right]}_{m}$ is the upper left $m×m$ block of the matrix $g$, and ${B}_{m}$ is the unit ball in the space of $m×m$ matrices over $K$ and proves that the image under ${\xi }_{m}$ of ${\sigma }_{n}$ is

${\sigma }_{n-m}\left(g\right)×\text{Const.det}{\left(1-{Z}^{*}Z\right)}^{r-1}dZ$

with $\tau =\frac{1}{2}\left(n-2m+1\right){dim}_{R}K$ and $dZ$ the Lebesgue measure. As an application the integrals

${\int }_{{U}^{0}\left(n,K\right)}\prod _{k=1}^{n}det{\left(1+{\left[g\right]}_{n-k+1}\right)}^{{\lambda }_{k}-{\lambda }_{k-1}}d{\sigma }_{n}\left(g\right)$

are evaluated in terms of the gamma function.

##### MSC:
 43A85 Analysis on homogeneous spaces 53C35 Symmetric spaces (differential geometry) 15A52 Random matrices (MSC2000)