This may be well-known but I couldn't find a way to charcterize the double-cosets of $U(n)\times U(n)$ in $U(2n)$ or couldn't find reference.


Is there reference where I can look for?



Double cosets of $U(n)\times U(n)$ in $U(2n)$
$K=U(n)\times U(n)$ is a symmetric subgroup of $G=U(2n)$. There is a discussion of $K$-double cosets in $G$ for any compact symmetric space in Helgason's book "Differential geometry, $\ldots$". See especially Thm. VII.8.6. That requires $G$ to be simply connected, though, but that is no problem since in your case $SU(2n)/K\_0=U(2n)/K$ where $K\_0=S(U(n)\times U(n))$.


The upshot of that discussion is that the Cartan decomposition $G=KAK$ holds where $A$ is a torus which othogonal to $K$ and is maximal with this property. Moreover, each $K$-double coset intersects $A$ is a $W\_0$-orbit where $W\_0$ is the restricted Weyl group of $G/K$. Thus $K\backslash G/K=A/W\_0$.


In your case $A=U(1)^n$ and $W\_0=S\_n\ltimes\mathbb Z\_2^n$ is the Weyl group of type $C\_n$ acting in the obvious way. 


