integral kernel function for the SU(N) group 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); 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
$$ dX_{SU(N)} = \frac{1}{N!(2\pi)^{N-1}} \det_{(N-1)X(N-1)}(K(\lambda_j,\lambda_k))_{1\leq j,k\leq N-1} \text{ } d\lambda_1\dots d\lambda_{N-1} $$ 
 A: 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, 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:

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.
