I am trying to understand the behavior of distributions over the Unitary group (i.e. the set of square matrices $P$ such that $P^tP = I_d$), or in general distribution over the Stiefel manifold (set of $D\times K$ matrices such that $P^tP = I_K$).
In particular, I need to understand how to compute any marginal distributions. Let's say for instance, the marginals of the uniform distribution of one the set defined above.
By Intuition I would like to say that spherical symmetries imply that the marginals are uniform on the corresponding Stiefel manifold, but I don't manage to write it properly...