Let \begin{align} (d\mathbf{H})= \bigwedge_{1\leq j\leq k\leq N} d h_{jj}^{(1)} \bigwedge_{1\leq j< k\leq N}d h_{jk}^{(2)} \ ... \bigwedge_{1\leq j< k\leq N}\ d h_{jk}^{(\beta)} \end{align} bet the volume measure of a symmetric/hermitian/self-dual matrix with $\beta=1/2/4$ (this implies that $h_{jk}$ is real/complex/real quaternion), where $(d\mathbf{H})$ is understood as the exterior product of all the independent elements of the matrix
\begin{equation} d\mathbf{H}= \begin{bmatrix} dh_{11}& dh_{12}&...&dh_{1N}\\dh_{21}&dh_{22}&...&dh_{2N}\\ .&.&.&.\\ dh_{N1}& dh_{N2}& ...& dh_{NN} \end{bmatrix}. \end{equation} If the set $\{h_{jk}\}$ is parametrized as $h_{jk} = a_{jk} g_{jk}$ with $a_{jk}$ a function of $\{g_{jk} \}$, how does this measure is transformed? This is a crucial point that I cannot understand and that is not explained in my ref. book (PJ Forrester, log gases and random matrices).
The result for $d\mathbf{H}$ real is stated in the book: \begin{equation} (d\mathbf{H})= \left[ \prod_{1\leq j\leq k\leq N} a_{jk} \right](d\mathbf{G}). \end{equation}
For the generalization to complex and real quaternion matrices, it would suit my purpose that the general formula be
\begin{equation} (d\mathbf{H})= \left[ \prod_{1\leq j\leq k\leq N} |a_{jk}|^\beta \right](d\mathbf{G}) \end{equation} but I cannot justify it! (nor find any other formulation). I'm not very familiar with this type of maths.
EDIT: I should add that the $a_{jk}$ are real in my case.
EDIT2: A pointer to some interesting litteratyre would be interesting! Forrester's reference on that point is closed source, unavailable at my university library and very old (hence not mordern at all in its notations, etc., as I could gather from the google book preview)
EDIT3: Clarified the question a lot, fixed typos and mistakes that I found on my own.
\begin{align} (d\mathbf{H})&= \bigwedge_{1\leq j\leq k\leq N} a_{jj} d g_{jj}^{(1)} \bigwedge_{1\leq j< k\leq N} a_{jk}d g_{jk}^{(2)} \ ... \bigwedge_{1\leq j< k\leq N}\ a_{jk}d g_{jk}^{(\beta)}\\ &= \prod_{1\leq j\leq k \leq N}( a_{jk})^\beta\ (d \mathbf{G}) \end{align}