Cuspidality of pullbacks of Siegel-Hilbert Eisenstein series on Hermitian symmetric domains

Abstract

Let $G_N$ be either a symplectic, unitary or Hermitian orthogonal group of rank $2N=2m+2n$ with $m\leq n$. We show that the restriction of a Siegel-Hilbert Eisenstein series on $G_N$ to the diagonally embedded group $G_m\times G_n$ has a nontrivial cuspidal component in the smaller variable. As a consequence, we explicitly construct classes of Siegel-Hilbert cuspforms with rational-valued Fourier coefficients.