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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 $U(n)$ to $U(n-1)$ for which the image of the normalized Haar measure of $U(n)$ is that of $U(n-1)$. 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 $H$, $U^0(n,K)$ is the group $SO(n)$, $U(n)$ or $Sp(n)$. The author considers the map $$\gamma^m: U^0(n,K)\to U^0 (n-m,K),$$ $$g=\left( \matrix P & Q\\ R & T\endmatrix \right)\mapsto T-R (1+P)^{-1}Q,$$ for an $(m+(n-m))\times (m+(n-m))$ block representation of $g$. It is proved that the image under the map $\gamma^m$ of the normalized Haar measure $\sigma_n$ of $U^0(n,K)$ is the Haar measure $\sigma_{n-m}$. Further the author considers the map$$\xi_m: U^0 (n,K) \to U^0(n-m,K) \times B_m,$$ $$g\mapsto \bigl(\gamma^m(g), [g]_m \bigr),$$ where $[g]_m$ is the upper left $m\times m$ block of the matrix $g$, and $B_m$ is the unit ball in the space of $m\times m$ matrices over $K$ and proves that the image under $\xi_m$ of $\sigma_n$ is $$\sigma_{n-m}(g)\times \text{Const.det} (1-Z^*Z)^{r-1}dZ$$ with $\tau={1\over 2}(n-2m+1)\dim_RK$ and $dZ$ the Lebesgue measure. As an application the integrals $$\int_{U^0(n,K)} \prod^n_{k=1} \det\bigl(1+[g]_{n-k+1} \bigr)^{\lambda_k-\lambda_{k-1}}d \sigma_n (g)$$ are evaluated in terms of the gamma function.

MSC:
43A85Analysis on homogeneous spaces
53C35Symmetric spaces (differential geometry)
15B52Random matrices
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References:
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