For $a=(a_1,a_2)$ and $b=(b_1,b_2)$ in $\mathbb{C}^2$ let me use the notation from classical invariant theory $$ (ab)=a_1b_2-a_2b_1\ . $$ If I run out of letters $a,b,c,\ldots$, then I will switch to $a^{(1)},a^{(2)},a^{(3)},\ldots$
Consider the following homogenized analogue of the complex Mehta integral $$ \int_{(\mathbb{C}^2)^n} e^{-\sum_{i=1}^n\sum_{\ell=1}^2|a^{(i)}_{\ell}|^2} \prod_{1\le i<j\le n}|(a^{(i)}a^{(j)})|^{2\gamma}\ \prod_{i=1}^n\prod_{\ell=1}^{2}\frac{d\Re a^{(i)}_{\ell}d\Im a^{(i)}_{\ell}}{\pi}\ . $$
Is there a simple explicit formula for it?
This was inspired from this other MO question
