# Fourier coefficients of Siegel Eisenstein series

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.

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!

• See a paper of Paul Feit from some decades ago? – paul garrett Aug 26 '19 at 23:06