# average Riemannian distance between Identiity and a random point in SO(n) or SU(n)

I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to evaluate the following integral:

$$\displaystyle Z_n^{-1} \int_{[-\pi,\pi]^n} (\sum_i f(\theta_i)^2)^{k/2} \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2 d \theta$$.

where $f(\theta) = |\theta|$ for $\theta \in [-\pi,\pi]$. But how does one rid of the square root? Alternatively one expresses $(\sum_i \theta_i^2)^k$ in terms of an infinite sum of power sum polynomials $p_j$, and then use results from Diaconis and Shahshahani on random matrices to show that the expectation of $p_j p_k$ satisfy an orthogonality relation. This is how I computed the even moments. But for odd ones one still needs to deal with square roots. Thanks in advance.

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