I am interested in the value of the following integral, for $n\ge 2$:
$$ \int_{\mathbb R^n} e^{-\frac 12 \lVert \lambda\rVert^2 + a_1 \lambda_1 + a_2 \lambda_2} \prod_{i < j} | \lambda_j - \lambda_i | \mathrm d \lambda_1 \dots \mathrm d \lambda_n $$
When taking $a_1=a_2=0$, this reduces to Mehta's integral with $\gamma=1/2$. This question is a special case of this related question on MO, which was not answered for $\gamma=1/2$. I am hoping that maybe 1) this case is simpler than the one proposed in the original question ; 2) seven years have passed and maybe someone has come up with an answer in-between!
I was hoping that maybe something like Aomoto's variant of Selberg's integral could be used to break the symmetry between the coefficients, but I cannot get it to work.
Any help would be appreciated!