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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)
##### MSC:
 33E20 Other functions defined by series and integrals 43A85 Harmonic analysis on homogeneous spaces
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