  Since recent X-ray observations have revealed that most clusters of galaxies
are surrounded by an X-ray emitting gaseous halo, it is reasonable to expect
that the Local Group of galaxies has its own X-ray halo. We show that such a
halo, with temperature $\sim 1$keV and column density $\sim O(10^{21}) {\rm
cm}^{-2}$, is a possible source for the excess low-energy component in the
X-ray background. The halo should also generate temperature anisotropies in the
microwave background via the Sunyaev-Zel'dovich effect. Assuming an isothermal
spherical halo with the above temperature and density, the amplitude of the
induced quadrupole turns out to be comparable to the COBE data without
violating the upper limit on the $y$-parameter. The induced dipole is
negligible compared to the peculiar velocity of the Local Group, and multipoles
higher than quadrupole are generally much smaller than the observed ones.
However non-sphericity and/or clumpiness of the halo will produce a stronger
effect. Therefore the gaseous halo of the Local Group, if it exists, will
affect the estimate of the primordial spectral index $n$ and the amplitude of
the density fluctuations deduced from the {\sl COBE} data.
