My question is about the derivation from Selberg integral to Dyson integral in this paper:
Selberg integral : $$ S_n(\alpha,\beta,\gamma) := \int_0 ^1 \cdots \int_0 ^1 \prod_{j=1}^n t_j^{\alpha-1}(1-t_j)^{\beta-1} \prod_{1\leqslant j<k\leqslant n} \lvert {t_j - t_k} \rvert ^{2\gamma} dt_1\cdots d t_n = \prod_{j=0}^{n-1} \frac{\Gamma (\alpha+j\gamma) \Gamma(\beta+j\gamma)\Gamma(1+(j+1)\gamma)} {\Gamma(\alpha+\beta+(n+j-1)\gamma)\Gamma(1+\gamma)}. $$
Dyson integral:
$$ C_n(\gamma) := \frac{1}{(2\pi)^n} \int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi} \prod_{1 \leqslant j<k \leqslant n} |e^{i \theta_j} - e^{i \theta_k}|^{2\gamma} d \theta_1 \cdots d\theta_n $$
Due to R. Askey,
the Selberg integral can be used to express the Dyson integral
directly.
Askey's observation is based on the easily established general
identity:
$$ \int_0^1 \cdots \int_0^1
(t_1\cdots t_n)^{z-1}
f(t_1,\dots,t_n) \, d t_1 \cdots d t_n \\
=\left( \frac{1}{2\sin \pi z} \right)^n
\int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi}
e^{iz(\theta_1+\cdots+\theta_n)}
f(- e^{i\theta_1} ,\dots, - e^{i\theta_n})
\, d\theta_1 \cdots d \theta_n, $$
which is valid for $f$ a Laurent polynomial and $\Re(z)$ large
enough so that the left-hand side exists.
Applying the identity to the Selberg integral with $\beta$ a positive integer
and $\gamma$ a nonnegative integer shows that
$$
S_n(\alpha,\beta,\gamma)=(-1)^{n+\binom{n}{2}\gamma}
\Bigl(\frac{\pi}{\sin \pi b}\Bigr)^n M_n(a,b,\gamma),
$$
where $\alpha:=-b-(n-1)\gamma,\ \ \beta:=a+b+1$ and
$$
M_n(a,b,\gamma) := \frac{1}{(2\pi)^n}
\int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi}
\prod_{j=1}^n e^{\frac{1}{2}i\theta_j (a-b)}
|{1+e^{i\theta_j}}|^{a+b} \\ \times
\prod_{1 \leqslant j<k \leqslant n}
|{e^{i \theta_j}-e^{i \theta_k}}|^{2\gamma} \,
d\theta_1 \cdots d \theta_n.
$$
From the Selberg integral,
the reflection formula and finally Carlson's theorem,
it follows that
$$
M_n(a,b,\gamma) =
\prod_{j=0}^{n-1}
\frac{\Gamma (1+a+b+j\gamma) \Gamma(1+(j+1)\gamma)}
{\Gamma (1+a+j\gamma)\Gamma (1+b+j\gamma) \Gamma (1+\gamma)},
$$
for $a,b,\gamma \in\mathbb C$ such that
$\Re (a+b+1)>0,\Re(\gamma)>-\min\{\frac1n, \frac{\Re(a+b+1)}{n-1}\}
$.
For $a=b=0$ this is the desired Dyson integral.
I have some questions about this calculation:
1) About $\alpha$, they identify $\alpha:=-b-(n-1)\gamma$. From my calculation, the term $-(n-1)\gamma$ in $\alpha$ comes from $\prod_{1 \leqslant j<k \leqslant n} |t_j - t_k|^{2\gamma} $ when changing variables from $t_r$ to $e^{i\theta_r}$, i.e. $$ t_j-t_k= e^{i\theta_j}-e^{i\theta_k}= |e^{i\theta_j}-e^{i\theta_k}| i e^{i\frac{\theta_j+\theta_k}{2}}.$$ If the Vandermonde product term in the Selberg integral is in the absolute value, there seems no such term $-(n-1)\gamma$ in $\alpha$. Anything wrong with my calculation?
2) Is the absolute value of the Vandermonde product necessary in the Selberg integral? What is the integral if there is no absolute value of the Vandermonde product?
