Abstract: The relative trace formula is a tool for studying automorphic representations
on symmetric spaces. In this work two relative trace formulas are studied: One
for GSp(2) relative to a form of SO(4) and one for PGL(n) relative to GL(n−1).
The first trace formula is used to study the Saito-Kurokawa lifting of automorphic
representations from PGL(2) to PGSp(2), thus characterising the
image as the representations with a nonzero period for the special orthogonal
group SO(4,E/F) associated to a quadratic extension E of the base field F,
and a nonzero Fourier coefficient for a generic character of the unipotent radical
of the Siegel parabolic subgroup.
The second trace formula is used to study the representations ! of PGL(n)
which have a nonzero linear form invariant under the Levi subgroup GL(n−1),
and a certain nonvanishing degenerate Fourier coefficient, over local and global
fields. The result is that these ! are of the form I(1n−2, !"), namely normalizedly
induced from the parabolic subgroup of type (n − 2, 2), trivial representation
on the GL(n − 2)-factor and cuspidal !" on the PGL(2)-factor. The analysis
of the formula is intricate our main achievement is rigorously establishing the
convergence of the formula. New in this case is that none of the representations
of PGL(n) involved are cuspidal, they rather occur in the continuous spectrum,
but discretely in our formula.