  We present a detailed formalism of the microscopic particle-rotor model for
hypernuclear low-lying states based on a covariant density functional theory.
In this method, the hypernuclear states are constructed by coupling a hyperon
to low-lying states of the core nucleus, which are described by the generator
coordinate method (GCM) with the particle number and angular momentum
projections. We apply this method to study in detail the low-lying spectrum of
$^{13}_{~\Lambda}$C and $^{21}_{~\Lambda}$Ne hypernuclei. We also briefly
discuss the structure of $^{155}_{~~\Lambda}$Sm as an example of heavy deformed
hypernuclei. It is shown that the low-lying excitation spectrum with positive
parity states of the hypernuclei, which are dominated by $\Lambda$ hyperon in
$s$-orbital coupled to the core states, are similar to that for the
corresponding core states, while the electric quadrupole transition strength,
$B(E2)$, from the 2$^+_1$ state to the ground state is reduced according to the
mass number of the hypernuclei. Our study indicates that the energy splitting
between the first 1/2$^-$ and 3/2$^-$ hypernuclear states is generally small
for all the hypernuclei which we study. However, their configurations depend
much on the properties of a core nucleus, in particular on the sign of
deformation parameter. That is, the first $1/2^-$ and $3/2^-$ states in
$^{13}_{~\Lambda}$C are dominated by a single configuration with $\Lambda$
particle in the $p$-wave orbits and thus providing good candidates for a study
of the $\Lambda$ spin-orbit splitting. On the other hand, those states in the
other hypernuclei exhibit a large configuration mixing and thus their energy
difference cannot be interpreted as the spin-orbit splitting for the
$p$-orbits.
