Poisson kernel for the orthogonal groups

fa.functional-analysisharmonic-analysislie-groupsrandom matricessymmetric-functions

For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\dagger Z\le1$ we have the Poisson kernel proportional to $|\det(X-Z)|^{-2N}$, which can be written in terms of Schur polynomials using the Cauchy identity for $\sum_\lambda s_\lambda(X)s_\lambda(Z)$.

For the real ball $z^2<1$ in $\mathbb{R}^n$, there is also a Poisson kernel, but this is proportional to $|x-z|^{-n}$. Just from analogy, I would imagine that this could be generalized to the (special?) orthogonal group $O(N)$ so that in the real matrix ball $Z^TZ\le1$ we would have a Poisson kernel proportional to $|\det(X-Z)|^{-N}$, which in turn could be written in terms of orthogonal characters using some version of the Cauchy identity.

However, I cannot find anything about this, neither in the literature about symmetric functions and Cauchy identities, nor in literature about analysis on matrix spaces. Why?

Best Answer

The Poisson kernel for the orthogonal group was calculated by Benjamin Béri in Generalization of the Poisson kernel to the superconducting random-matrix ensembles. This is in the context where $X$ is the scattering matrix of a superconductor in the Cartan symmetry class D (broken time-reversal and spin-rotation symmetries).

The result for $X\in{\rm SO}(N)$ (real unitary matrices with determinant $+1$) is $$P_{\rm SO}(X)\propto\frac{1}{|\operatorname{det}(X-Z)|^{N-1}},$$ to be contrasted with $$P_{\rm U}(X)\propto\frac{1}{|\operatorname{det}(X-Z)|^{2N}}$$ for $X\in{\rm U}(N)$.

Since the exponent $N-1$ differs from the expectation in the OP, let me check the reproducing property for ${\rm SO}(2)$ and $Z=zI_2$ (with $-1<z<1$). In that case the Poisson kernel is $$P_{\rm SO}(X)d\mu(X)=f(\theta)\,d\theta,\;\;f(\theta)=\frac{1-z^2}{2 \pi \left(z^2-2 z \cos \theta+1\right)}, $$ $$\text{in the parameterization}\;\;X(\theta)=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}\;\;0<\theta<2\pi,$$ and one readily checks that $$\langle X^p\rangle\equiv\int_0^{2\pi} X^p(\theta)f(\theta)\,d\theta=z^p I_2=Z^p.$$ The reproducing property for the orthogonal Poisson kernel fails if $Z$ is not proportional to the unit matrix, for example, if $Z={{z\;0}\choose{0\,-z}}$ one has $\operatorname{det}(X-Z)=1-z^2$, independent of $X$ and $\langle X\rangle=0\neq Z$.

To generalize the Poisson kernel to all three classical groups ${\rm U}(N)$, ${\rm SO}(N)$, ${\rm Sp}(N)$, one defines an $N\times N$ subunitary matrix $Z$ as the submatrix of the $2N\times 2N$ unitary matrix $$\Omega=\begin{pmatrix}Z&T'\\ T&Z'\end{pmatrix}.$$ The $N\times N$ unitary matrix $X$ is then constructed by $$X=Z+T'X_0(1-Z'X_0)^{-1}T,$$ with $X_0$ an $N\times N$ unitary matrix distributed according to the Haar measure on the unitary, orthogonal, or symplectic group. The Poisson kernel is then defined as the corresponding probability distribution function $P(X)$ for a given $Z$. The result is $P(X)\propto|\operatorname{det}(X-Z)|^{-\beta N+\gamma}$, with $\beta=2,\gamma=0$ for ${\rm U}(N)$, $\beta=1,\gamma=1$ for ${\rm SO}(N)$, and $\beta=1,\gamma=-1$ for ${\rm Sp}(N)$. $$\mbox{}$$ The reproducing property, $\int P(X)f(X)d\mu(X)=f(Z)$, is ensured if the average of the matrix product $X_0(Z'X_0)^p$ vanishes for all $p=0,1,2\ldots$, which happens for any integer $N$ and any $Z$ in the unitary group, and for even $N$ and $Z\propto I_N$ in the orthogonal or symplectic groups.
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