Itzykson-Zuber integral over orthogonal groups I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case.
$I=\int_{\mathcal{O}(p)} \exp\left(-\frac{1}{2}~\mathrm{tr}(\Sigma^{-1}HLH^{T})\right)~\mathrm{d}H$
where ${\mathcal{O}(p)}$ is a Orthogonal group of $p \times p$ symmetric matrices, $\Sigma \neq aI $ is a covariance matrix of a column vector of a correlated Gaussian matrix $A$ that occurs in the Wishart matrix formation $W=A^{T}A$, $dH$ is normalized Harr measure and $L$ is the diagonal matrix $\mathrm{diag}(l_1,l_2,\cdots,l_p)$
I know that an infinite Zonal polynomial series exists for the integral. But, I am interested to know is there any closed-form or  tractable solutions for the integral.
 A: I present the answer for $p=2$ in here. We write the orthogonal matrix $H$ by using the Cayley' parametrization:
\begin{equation}
H = \left(1 + A\right) \cdot \left( 1 - A \right)^{-1}
\end{equation}
where 
\begin{equation}
A = \left(
\begin{array}{cc} 0 & a \\ -a & 0 \end{array}
\right)
\end{equation}
is a real skew-symmetric matrix. A straightforward calculation shows that:
\begin{equation}
H = \frac{1}{1+a^2} \left(
\begin{array}{rr}
1-a^2 & 2 a \\ - 2 a & 1- a^2
\end{array}
\right)
\end{equation}
Now we have:
\begin{equation}
\left(\det \Sigma\right) Tr\left[\Sigma^{-1} H L H^T\right] =
\frac{1}{2} Tr[\Sigma](l_1+l_2)
+
\frac{(l_1-l_2)}{2}\left((\Sigma_{1,1} - \Sigma_{2,2}) \cos(4 y) - 2 \Sigma_{1,2}\sin(4 y)\right) 
\end{equation}
where $a := \tan(y)$. Another straightforward calculation gives the Jacobian:
\begin{equation}
\frac{\partial(c_{1,1}, c_{1,2}, c_{2,2})}{\partial(l_1,l_2,a)}
= \frac{2 (l_1-l_2)}{1+a^2}
\end{equation}
where
\begin{equation}
\left(
\begin{array}{cc} c_{1,1} & c_{1,2} \\ c_{1,2} & c_{2,2} \end{array}
\right)
=
H L H^T
\end{equation}
Bringing everything together our integral reads:
\begin{eqnarray}
I &=& 2 (l_1-l_2) e^{-\frac{1}{4} \frac{Tr\left[\Sigma\right]}{\det \Sigma} (l_1+l_2)} \cdot
\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
e^{\frac{(l_2-l_1)}{4 \det \Sigma} \left((\Sigma_{1,1}-\Sigma_{2,2}) \cos(4 y) - 2 \Sigma_{1,2} \sin(4 y)\right)} d y \\
&=&  (l_1-l_2) e^{-\frac{1}{4} \frac{Tr\left[\Sigma\right]}{\det \Sigma} (l_1+l_2)} \cdot I_0\left((l_2-l_1) \frac{\sqrt{(Tr \Sigma)^2 - 4 \det(\Sigma)}}{4 \det \Sigma}\right)
\end{eqnarray}
where $I_0$ is the modified Bessel function of order zero.  As we can see the results depends only on the rotational invariants of the matrix $\Sigma$.
A: Take a look in the  abstracts of the 16 workshop in Bedlew (non coummutativ harmonic..) 6-12 of 7
