  We consider a planar heterostructure consisting of a normal metal in
proximity to an antiferromagnetic insulator, with an interlayer exchange
coupling between the metal and the insulator. The coupling to the two
sublattices of the antiferromagnetic insulator is allowed to be asymmetric. An
effective electron-electron interaction in the normal metal, mediated by
antiferromagnetic magnons in the insulator, is derived to second order in the
interlayer exchange coupling. Particular emphasis is placed on including
analytically derived expressions for the effective interactions including
Umklapp processes in the solutions to the superconducting gap equation. The gap
equation is first solved at the critical temperature as an eigenvalue problem
by linearizing the gap equation. The eigenvectors yield information on the
symmetry of the superconducting gap at the onset of superconductivity, and we
derive a phase diagram for the order parameter in this case. In the various
regimes of the phase diagram, we find $p$-wave, $f$-wave, and $d$-wave
superconductivity, with $p$-wave superconductivity in the dominant part of the
phase diagram. Umklapp processes, that come into play with increasing size of
the Fermi surface, yield $f$- and $d$-wave symmetries as the preferred
symmetries when band filling approaches half filling. To investigate the
stability of this order parameter symmetry as the temperature is lowered, we
also consider the nonlinear gap equation at zero temperature. We conclude that
the phase diagram and the symmetries of the superconducting order parameter
essentially are left intact as the temperature is lowered to zero temperature.
