TITLE: About taking an expectation over orthogonal matrices
QUESTION [6 upvotes]: 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 $Q \sim SO(m)$

REPLY [8 votes]: 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.