Primitive Fourier Coefficients of Hermitian Klingen-Eisenstein Series

Source abstract

Let $E/F$ be a CM extension and let $1\le r<n$. We determine primitive Fourier coefficients of Hermitian Klingen-Eisenstein series on $\mathrm{U}_{n,n}$ induced from cusp forms on $\mathrm{U}_{r,r}$, at principal congruence level. Using the pullback formula for Hermitian Eisenstein series, we express a normalized primitive Fourier coefficient of a Klingen-Eisenstein series as the corresponding primitive Fourier coefficient of a Siegel Eisenstein series multiplied by a Petersson inner product with an explicitly constructed holomorphic automorphic form. The automorphic form in the Petersson inner product is a finite sum of products of Hermitian theta series and Siegel Eisenstein series of degree $r$. Under some hypotheses, we also prove the $p$-integrality of these normalized primitive Fourier coefficients.