  We investigate the 1/3 monolayer $\alpha$-Pb/Si(111) surface by scanning
tunneling spectroscopy (STS) and fully relativistic first-principles
calculations. We study both the high-temperature $\sqrt{3}\times\sqrt{3}$ and
low-temperature $3\times 3$ reconstructions and show that, in both phases, the
spin-orbit interaction leads to an energy splitting as large as $25\%$ of the
valence-band bandwidth. Relativistic effects, electronic correlations and
Pb-substrate interaction cooperate to stabilize a correlated low-temperature
paramagnetic phase with well-developed lower and upper Hubbard bands coexisting
with $3\times3$ periodicity. By comparing the Fourier transform of STS
conductance maps at the Fermi level with calculated quasiparticle interference
from non-magnetic impurities, we demonstrate the occurrence of two large
hexagonal Fermi sheets with in-plane spin polarizations and opposite
helicities.
