  We investigate the electron-positron pair cascade taking place in the
magnetosphere of a rapidly rotating black hole. Because of the spacetime frame
dragging, the Goldreich-Julian charge density changes sign in the vicinity of
the event horizon, which leads to an occurrence of a magnetic-field aligned
electric field, in the same way as the pulsar outer-magnetospheric accelerator.
In this lepton accelerator, electrons and positrons are accelerated in the
opposite directions, to emit copious gamma-rays via the curvature and
inverse-Compton processes. We examine a stationary pair cascade, and show that
a stellar-mass black hole moving in a gaseous cloud can emit a detectable
very-high-energy flux, provided that the black hole is extremely rotating and
that the distance is less than about 1 kpc. We argue that the gamma-ray image
will have a point-like morphology, and demonstrate that their gamma-ray spectra
have a broad peak around 0.01-1 GeV and a sharp peak around 0.1 TeV, that the
accelerators become most luminous when the mass accretion rate becomes about
0.01% of the Eddington rate, and that the predicted gamma-ray flux little
changes in a wide range of magnetospheric currents. An implication of the
stability of such a stationary gap is discussed.
