Hua-type integrals over unitary groups and over projective limits of unitary groups
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 \[ \gamma^m: U^0(n,K)\to U^0 (n-m,K), \] \[ g=\left( \begin{matrix} P & Q\\ R & T\end{matrix} \right)\mapsto T-R (1+P)^{-1}Q, \] 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.
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