  We study the dynamics of merging compact binaries near a rotating
supermassive black hole (SMBH) in a hierarchical triple configuration. We
include various general relativistic effects that couple the inner orbit, the
outer orbit and the spin of the SMBH. During the orbital decay due to
gravitational radiation, the inner binary can encounter an "apsidal precession
resonance" and experience eccentricity excitation. This resonance occurs when
the apsidal precession rate of the inner binary matches that of the outer
binary, with the precessions driven by both Newtonian interactions and various
post-Newtonian effects. The eccentricity excitation requires the outer orbit to
have a finite eccentricity, and is most effective for triples with small mutual
inclinations, in contrast to the well-studied Lidov-Kozai effect. The resonance
and the associated eccentricity growth may occur while the binary emits
gravitational waves in the low-frequency band, and may be detectable by future
space-based gravitational wave detectors.
