  It has been suggested that there is possibly a class of stellar-mass black
holes (BHs) residing near (distance $\le 10^3 M$) the galactic center massive
black hole, Sgr A*. Possible formation scenarios include the mass segregation
of massive stellar-mass black holes and/or the disk migration if there was an
active accretion flow near Sgr A* within $\mathcal{O}(10)$ Myr. In this work,
we explore the application of this type of objects as sources of space-borne
gravitational wave detectors, such as Laser Interferometer Space Antenna
(LISA). We find it is possible to probe the spin of Sgr A* based on the
precession of the orbital planes of these stellar-mass black holes moving
around Sgr A*. We also show that the dynamical friction produced by accumulated
cold dark matter near Sgr A* generally produces small measurable phase shift in
the gravitational waveform. In the case that there is an axion cloud near Sgr
A*, the dynamical friction induced modification to gravitational waveform is
measurable only if the mass of the axion field is in a narrow range of the mass
spectrum. Gravitational interaction between the axion cloud and the
stellar-mass black holes may introduce additional precession around the spin of
Sgr A*. This additional precession rate is generally weaker than the
spin-induced Lense-Thirring precession rate, but nevertheless may contaminate
the spin measurement in a certain parameter regime. At last, we point out that
the multi-body gravitational interaction between these stellar-mass black holes
generally causes negligible phase shift during the LISA lifetime.
