  Recent X-ray observations have revealed remarkable correlations between the
masses of central supermassive black holes (SMBHs) and the X-ray properties of
the hot atmospheres permeating their host galaxies, thereby indicating the
crucial role of the atmospheric gas in tracing SMBH growth in the high-mass
regime. We examine this topic theoretically using the IllustrisTNG cosmological
simulations and provide insights to the nature of this SMBH -- gaseous halo
connection. By carrying out a mock X-ray analysis for a mass-selected sample of
TNG100 simulated galaxies at $z=0$, we inspect the relationship between the
masses of SMBHs and the hot gas temperatures and luminosities at various
spatial and halo scales -- from galactic ($\sim R_{\rm e}$) to group/cluster
scales ($\sim R_{\rm 500c}$). We find strong SMBH-X-ray correlations mostly in
quenched galaxies and find that the correlations become stronger and tighter at
larger radii. Critically, the X-ray temperature ($k_{\rm B}T_{\rm X}$) at large
radii ($r\gtrsim5R_{\rm e}$) traces the SMBH mass with a remarkably small
scatter ($\sim0.2$ dex). The relations emerging from IllustrisTNG are broadly
consistent with those obtained from recent X-ray observations. Overall, our
analysis suggests that, within the framework of IllustrisTNG, the present-time
$M_{\rm BH}-k_{\rm B}T_{X}$ correlations at the high-mass end ($M_{\rm
BH}\gtrsim 10^8 M_\odot$) are fundamentally a reflection of the SMBH mass --
halo mass relation, which at such high masses is set by the hierarchical
assembly of structures. The exact form, locus, and scatter of those scaling
relations are, however, sensitive to feedback processes such as those driven by
star formation and SMBH activity.
