  An electron-positron cascade in the magnetospheres of Kerr Black Holes (BH)
is a fundamental ingredient to fueling the relativistic $\gamma$-ray jets seen
at the polar regions of galactic supermassive BHs (SMBH). This leptonic cascade
occurs in the "spark gap" region of a BH magnetosphere where the unscreen
electric field parallel to the magnetic field is present, hence it is affected
by the magnetic field structure. A previous study explored the case of a thin
accretion disk, representative of Active Galactic Nuclei (AGN). Here we explore
the case of a quasi-spherical gas distribution, as is expected to be present
around the SMBH Sgr A* in the center of our Milky Way galaxy, for example. The
properties and efficiency of the leptonic cascade are studied. The findings of
our study and the implications for SMBH systems in various spectral and
accretion states are discussed. The relationships and scalings derived from
varying the mass of the BH and background photon spectra are further used to
analyze the leptonic cascade process to power jets seen in astronomical
observations. In particular, one finds the efficiency of the cascade in a
quasi-spherical gas distribution peaks at the jet axis. Observationally, this
should lead to a more prominent jet core, in contrast to the thin disk
accretion case, where it peaks around the jet-disk interface. One also finds
the spectrum of the background photons to play a key role. The cascade
efficiency is maximum for a spectral index of two, while harder and softer
spectra lead to a less efficient cascade.
