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Let GN be either a symplectic, unitary or Hermitian orthogonal group of rank 2N = 2m+2n with m ≤ n. We show that the restriction of a Siegel-Hilbert Eisenstein series on GN to the diagonally embedded group Gm×Gn 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.

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