Suppose that we have a uniformly distributed $d\times d$ random orthonormal matrix $\mathbf{X}$. Here "uniform" is defined in the sense of Haar measure, i.e., the distribution does not change up to any rotation of basis (e.g., multiplication with any arbitrary orthonormal matrix).

Let $\mathbf{Z}$ be the upper-left $k\times k$ principal submatrix of $\mathbf{X}$. What is the distribution of $\| \mathbf{Z} \|_F^2$, e.g., its probability density function and cumulative distribution function? (We may be also interested in their asymptotic versions.)

This also relates to the so-called multivariate beta distribution. Let $\mathbf{Y}\in \mathbb{R}^{d\times k}$ be a random matrix with i.i.d. standard Gaussian entires, and $\mathbf{Y} = \mathbf{Q}\mathbf{R}$ be its QR decomposition. Let $\mathbf{Q}'$ be the top $k\times k$ submatrix of $\mathbf{Q}$. Then $\mathbf{Q}'$ follows the multivariate beta distribution. An equivalent question is as follows. What is the (asymptotic) distribution of $\|\mathbf{Q}'\|_F^2$?