It is well know that the Haar probability measure for the $U(N)$ group, given by
$$
\begin{align}
dX\_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
 1 & 1 & \cdots & 1 & 1 \\
 e^{i\lambda\_1} & e^{i\lambda\_2} & \cdots & e^{i\lambda\_{N-1}} & e^{i\lambda\_N} \\
 e^{i2\lambda\_1} & e^{i2\lambda\_2} & \cdots & e^{i2\lambda\_{N-1}} & e^{i2\lambda\_N} \\
 \vdots & \vdots & \ddots & \vdots & \vdots \\
 e^{i(N-1)\lambda\_1} & e^{i(N-1)\lambda\_2} & \cdots & e^{i(N-1)\lambda\_{N-1}} & e^{i(N-1)\lambda\_N} \\
\end{vmatrix}^2 d\lambda\_1\dots d\lambda\_N \\
 & = \frac{1}{N!(2\pi)^N} \prod\_{1\leq j<k\leq N} |e^{i\lambda\_j}-e^{i\lambda\_k}|^2 d\lambda\_1\dots d\lambda\_N 
\end{align}
$$
can also be expressed as a determinantal point process
$$ dX\_{U(N)} = \frac{1}{N!(2\pi)^N} \det\_{NXN}(S\_N(\lambda\_j,\lambda\_k))\_{1\leq j,k\leq N} \text{ } d\lambda\_1\dots d\lambda\_N $$ 
where 
$$S\_N(x,y)=\frac{\sin\frac{N(x-y)}{2}}{\sin\frac{x-y}{2}}$$
is the integral kernel function for the $U(N)$ group (a proof of this fact can be found, for example, in section 4.1 of [this paper](https://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/handouts/conrey/Conrey_NotesEValueDistrClassicalCompGrps.pdf)); similar kernels have also been found for the special orthogonal and symplectic groups.


My question is whether any analogous kernel function is know for the $SU(N)$ group ? (or if it can be proved that it doesn't exist).


If such a function $K(x,y)$ exists, it would allow one to express the $SU(N)$ Haar probability measure
$$
\begin{align}
dX\_{SU(N)} & = \frac{1}{N!(2\pi)^{N-1}}
\begin{vmatrix}
 1 & \cdots & 1 & 1 \\
 e^{i\lambda\_1} & \cdots & e^{i\lambda\_{N-1}} & e^{-i(\lambda\_1+\ldots+\lambda\_{N-1})} \\
 e^{i2\lambda\_1} & \cdots & e^{i2\lambda\_{N-1}} & e^{-i2(\lambda\_1+\ldots+\lambda\_{N-1})} \\
 \vdots & \ddots & \vdots & \vdots \\
 e^{i(N-1)\lambda\_1} & \cdots & e^{i(N-1)\lambda\_{N-1}} & e^{-i(N-1)(\lambda\_1+\ldots+\lambda\_{N-1})} \\
\end{vmatrix}^2 d\lambda\_1\dots d\lambda\_{N-1} \\
 & = \frac{1}{N!(2\pi)^{N-1}} \prod\_{1\leq j<k\leq N-1} |e^{i\lambda\_j}-e^{i\lambda\_k}|^2 \prod\_{1\leq j\leq N-1} |e^{i\lambda\_j}-e^{-i(\lambda\_1+\ldots+\lambda\_{N-1})}|^2 d\lambda\_1\dots d\lambda\_{N-1} \\
 % & = P(\lambda\_1,\dots,\lambda\_{N-1}) d\lambda\_1\dots d\lambda\_{N-1}
\end{align}
$$
in the form of
$$ dX\_{SU(N)} = \frac{1}{N!(2\pi)^{N-1}} \det\_{(N-1)X(N-1)}(K(\lambda\_j,\lambda\_k))\_{1\leq j,k\leq N-1} \text{ } d\lambda\_1\dots d\lambda\_{N-1} $$ 



integral kernel function for the SU(N) group
The integral kernel for ${\rm U}\,(N)$, due to Dyson, has been generalized by Katz and Sarnak to other compact groups ([Random Matrices, Frobenius Eigenvalues, and Monodromy](https://web.math.princeton.edu/~nmk/RMFEM.pdf), page 121). Their result has the general form
$$d\mu=\frac{1}{n!}\det\_{n\times n}[L\_N(\lambda\_i,\lambda\_j)]\prod\_{i=1}^{n}\frac{d\lambda\_i}{\sigma\pi},\;\;\lambda\_i\in[0,\sigma\pi],\;\;1\leq i\leq n,$$
$$S\_N(x)=\frac{\sin(Nx/2)}{\sin(x/2)},\;\;
L\_N(x,y)=\tfrac{1}{2}\sigma[S\_{\rho N+\tau}(x-y)+\varepsilon S\_{\rho N+\tau}(x+y)].$$
The coefficients are tabulated as follows:


![](https://ilorentz.org/beenakker/MO/KatzSarnak.png)


Note that $\mu[{\rm O}\_-(2N+1)]$ is the same as $\mu[{\rm SO}(2N+1)]$, since the matrices differ by a minus sign. Also note that $\mu[{\rm O}\_-(2N+2)]=\mu({\rm U\,Sp}(2N)]$.


The group ${\rm SU}\,(N)$ is conspicuously missing from this table... I would assume there is a reason for this (Katz and Sarnak discuss ${\rm SU}\,(N)$ at various other points in their text). My surmise is that there is no way to incorporate the delta function $\delta\bigl(\sum\_{i=1}^N \lambda\_i\bigr)$ into an $(N-1)\times(N-1)$ determinant.


