I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--never mind if it is well defined:
$$ \int A^\dagger d A $$
where A is some $N^2 \times N^2$ unitary matrix. I would like to know how this integral looks after making the transformation to coordinates given by $U,\{\lambda\}$
$$ A_{ab}=A_{(iN+j)b}=\sum_\xi \lambda_\xi U_{i\xi b} U_{\xi j b} \quad i,j \in [0,N-1] $$ where here we have the nice, but perhaps useless, orthogonality property that: $\sum_j U^*_{ij b}U_{jk b}=\delta_{ik}$.
I have been struggling for an embarrassingly long time on (a slightly more complicated version of-) this, and for whatever reason I am failing to compute the Jacobian and the form of the integral. Can anyone help? Thanks!