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 $B$. 




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



About taking an expectation over orthogonal matrices
Let's denote $S=\{1,2,\dots m\}$ and $A\subset S$ with $|A|=k$. Then from orthogonality of $Q$ we have 
$$Q\_{A,S}Q\_{A,S}^{T}=I\_k$$ therefore Cauchy Binet tells us that
$$\sum\_{B\in \binom{S}{k}}\det(Q\_{A,B})^2=1.$$
If you apply expectation on both sides and notice that $E(\det(Q\_{A,B})^2)$ doesn't depend on $B$ by invariance of the Haar measure you get your identity.


