  Experimentally observed ground state band based on the $1/2^{-}[521]$ Nilsson
state and the first exited band based on the $7/2^{-}[514]$ Nilsson state in
the odd-$Z$ nucleus $^{255}$Lr are studied by the cranked shell model (CSM)
with the paring correlations treated by the particle-number-conserving (PNC)
method. This is the first time the detailed theoretical investigations being
performed on these rotational bands. Both the experimental kinematic and
dynamic moment of inertia ($\mathcal{J}^{(1)}$ and $\mathcal{J}^{(2)}$) versus
rotational frequency are reproduced quite well by the PNC-CSM calculations. By
comparing the theoretical kinematic moment of inertia $\mathcal{J}^{(1)}$ with
the experimental ones extracted from different spin assignments, the spin
$17/2^{-}\rightarrow13/2^{-}$ is assigned to the lowest-lying $196.6(5)$ keV
transition of the $1/2^{-}[521]$ band, and $15/2^{-}\rightarrow11/2^{-}$ to the
$189(1)$ keV transition of the $7/2^{-}[514]$ band, respectively. The proton
$N=7$ major shell is included in the calculations. The intruder of the high$-j$
low$-\Omega$ orbitals $1j_{15/2}$ $ (1/2^{-}[770])$ at the high spin leads to
the band-crossing at $\hbar\omega\approx0.20$ ($\hbar\omega\approx0.25$) MeV
for the $7/2^{-}[514]$ $\alpha=-1/2$ ($\alpha=+1/2$) band, and at
$\hbar\omega\approx0.175$ MeV for the $1/2^{-}[521]$ $\alpha=-1/2$ band,
respectively. Further investigations show that the band-crossing frequencies
are quadrupole deformation dependent.
