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Application of the Minlos and Poincaré theorems to the asymptotic study of an orbital integral. (Application des théorèmes de Minlos et Poincaré à l’étude asymptotique d’une intégrale orbitale.) (French. English summary) Zbl 1149.33004
The author studies the asymptotic behaviour of the Otzykson-Zuber integral \( I_n(x^{(n),y^{(n)}})=\int_{K_n}e^{-i tr (x^{(n)}u y^{(n)}u^*)}\alpha^{(n)}(du) \), where \( x^{(n)}, y^{(n)} \) are hermitian \( n\times n\) matrices with coefficients in \( \mathbb F=\mathbb R,\mathbb C, \mathbb H\), \( K_n \) is the group of unitary matrices with coefficients in \( \mathbb F \) and \( \alpha^{(n)} \) is the normalized Haar measure of the group \( K_n \). Because of the invariance of the Haar measure \( \alpha^{(n)} \), it is enough to consider diagonal matrices. Here \( x^{(n)}= \text{diag}(\lambda_1,\cdots, \lambda_m,0,0,\cdots,0)\) with fixed values \( \lambda_j \) (for some fixed \( m \)) and \( y^{(n)}= \text{diag}(a_1^{(n)},\cdots,a_n^{(n)})\). The author proves the following theorem:
Suppose that for all k, \( \lim_{n\to \infty}\frac{a_k^{(n)}}{n}=\alpha_k \), then the sequence \( \alpha=(\alpha_k) \) is summable, such that
\[ \lim_{n\to \infty}\sum_{k=1}^n\frac{a_k^{(n)}}{n} =\sum_{k=1}^\infty \alpha_k \] and
\[ \lim_{n\to \infty}\sum_{k=1}^1\left (\frac{a_k^{(n)}}{n}\right)^2 =\sum_{k=1}^\infty \alpha_k^2 . \] Then
\[ \lim_{n\to\infty }I_n(x,y^{(n)})=\prod _{j=1}^m \varphi(\lambda_ j), \] where \( \varphi(\lambda)=\prod_{k=1}^\infty (1+i \frac{2}{d }\lambda \alpha_k)^{-\frac{d}{2}} \) (\( d=\dim_{\mathbb R}\mathbb F)\) and the convergence is uniform on each compact subset of \( \text{Herm}(m,\mathbb F) \). The main interest of the paper is the method used in the proof. One sees first that \(I_n(x,y)\) can be written as an integral on a Stiefel manifold in the space \(M(m,n;{\mathbb F})\) of \(m\times n\) matrices with entries in \(\mathbb F\). By a generalized Poincaré theorem, the normalized uniform measure on this manifold converges to a Gaussian measure on the infinite-dimensional space \(M(m,\infty;{\mathbb F})\) as \(n\) goes to infinity. Then, by using the Minlos theorem, it is shown that the integral \(I_n(x,y^{(n)})\) converges to an integral with respect to this Gaussian measure.
Reviewer: Jean Ludwig (Metz)
33E20 Other functions defined by series and integrals
43A85 Harmonic analysis on homogeneous spaces
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