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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!

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    $\begingroup$ See a paper of Paul Feit from some decades ago? $\endgroup$ – paul garrett Aug 26 '19 at 23:06
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This is dealt with to some extent in "Green forms and the arithmetic Siegel–Weil formula" by Garcia and Sankaran, see Section 5.2.2. Indeed the coefficients can be expressed in terms of Whittaker integrals of lower rank and Eisenstein series for GL_r.

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