| We apply some ideas of Bombieri and Garrett to construct natural self-adjoint operators on spaces of automorphic forms whose only possible discrete spectrum is lambdas = s(s --- 1) for s in a subset of on-line zeros of an L-function, appearing as a compact period of cuspidal-data Eisenstein series on GL 4. These ideas have their origins in results of Hejhal and Colin de Verdiere. In parallel with the GL(2) case, the corresponding pair-correlation and triple-correlation results limit the fraction of on-the-line zeros that can appear in this fashion. |
