Say $Q$ is a random variable which is sampling orthogonal matrices in $m$ dimensions using the Haar measure on $O(m)$. Let $A$ and $B$ be some (fixed) subset of rows and columns of $Q$ such that $\vert A \vert = \vert B \vert = k$.

Now is such an identity true? (If yes then could you kindly give the proof or a reference for it!)

$\mathbb{E}_{Q \sim O(m)}[det^2(Q_{A,B})] = \frac {1}{\binom {m}{k} }$

where by $Q_{A,B}$ we mean the submatrix of $Q$ corresponding to the rows from $A$ and columns from $B$.

If it seems necessary for the proof feel free to assume that $Q \sim SO(m)$