I am looking for reference about Fourier coefficients of Eisenstein series. Currently I am mainly interested Eisenstein series given by Siegel parabolic subgroup of $SP_{2n}$ and $U(n,n)$. Let's consider symplectic case for now. Given a symmetric matrix T, we can define a Fourier coefficient with respect to T:

$E_T(g,s,\Phi)$=$\int_{[N(A)]}$$E(n(b)g,s,\Phi )$$\psi (-tr(Tb)) db$.

If $det(T)\neq 0$, then by unfolding, only one term is nonzero, which is $\int_{sym(A)}\Phi(\omega^{-1}n(b)g,s)\psi(-tr(Tb))db$ where $\omega$ is:$$\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \\ \end{pmatrix}$$ Then if $\Phi$ is factorizable, we can decompose the Fourier coefficient into product of Whittaker integral to study the Fourier Coefficient. Further study will be about local density, Siegel-Weil formula, Arithmetic Siegel-Weil formula etc.

But if T is singular, then more terms can survive. I guess we can relate the terms with sum of nonsingular Fourier coefficients of Eisenstein series comes from $SP_{2r}$ where $r<n$.

So is there any reference that deal with this kind of singular Fourier coefficient of Eisenstein series? I tried to search, but found it is very hard to get useful result by just googling. Thank you!