  X-ray emission (XES) and absorption (XAS) spectra near the B-$K$ edge were
measured on single-crystalline AlB$_2$ compound which is an isostructural
diboride of superconducting MgB$_2$. The partial density of states (PDOS) of
B-2$p\sigma$ and $p\pi$ orbitals were derived from the polarization dependence
of XES and XAS spectra. There are considerable amounts of PDOS near the Fermi
energy in AlB$_2$ similarly to that in MgB$_2$, but there are almost no PDOS in
$p\sigma$ orbitals of AlB$_2$ near the Fermi energy, i.e., a pseudo-gap in
$p\sigma$ state and a broad metallic state in $p\pi$ state are observed. The
present result indirectly supports scenarios that the $p\sigma$ holes play an
important role in the occurrence of superconductivity in MgB$_2$. The overall
features of PDOS were found to be in good agreement with the result of band
calculation of AlB$_2$, but a small discrepancy in the Fermi energy is
observed, which is attributed to the Al vacancy in the compounds, i.e., the
estimated concentration is Al$_{0.93}$B$_2$.
