I was looking at a 0-dimensional matrix model, where the variables are $N\cdot N$ Hermitean matrices. It had a gauge symmetry, e.g. $U(N)$. And in the path integral, the Faddeev-Popov trick was used. So instead of integrating over the set of matrices, one was reduced to integrating over the $N$ (real) eigenvalues. Or, I tried counting the number of degrees of freedom and I was puzzled:
N by N Hermitean matrices have $N^2$ real independent variables ($N(N-1)$ values for the triangular part (non counting the diagonal ones), plus $N$ real variables for the diagonal ones, giving us a total of $N^2$
$U(N)$ has real dimension $N^2$, classic result from group theory
At the end, I arrive at $N^2 - N^2 = 0$ variables instead of $N$ real eigenvalues.
I tried seeing it from another point but I can't derive what I'd like to. What I am thinking/doing wrong?