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We present a detailed formalism of the microscopic particle-rotor model forhypernuclear low-lying states based on a covariant density functional theory.In this method, the hypernuclear states are constructed by coupling a hyperonto low-lying states of the core nucleus, which are described by the generatorcoordinate method (GCM) with the particle number and angular momentumprojections. We apply this method to study in detail the low-lying spectrum of$^{13}_{~\Lambda}$C and $^{21}_{~\Lambda}$Ne hypernuclei. We also brieflydiscuss the structure of $^{155}_{~~\Lambda}$Sm as an example of heavy deformedhypernuclei. It is shown that the low-lying excitation spectrum with positiveparity states of the hypernuclei, which are dominated by $\Lambda$ hyperon in$s$-orbital coupled to the core states, are similar to that for thecorresponding core states, while the electric quadrupole transition strength,$B(E2)$, from the 2$^+_1$ state to the ground state is reduced according to themass number of the hypernuclei. Our study indicates that the energy splittingbetween the first 1/2$^-$ and 3/2$^-$ hypernuclear states is generally smallfor all the hypernuclei which we study. However, their configurations dependmuch on the properties of a core nucleus, in particular on the sign ofdeformation 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 studyof the $\Lambda$ spin-orbit splitting. On the other hand, those states in theother hypernuclei exhibit a large configuration mixing and thus their energydifference cannot be interpreted as the spin-orbit splitting for the$p$-orbits.

Microscopic study of low-lying spectra of $\Lambda$ hypernuclei based on a beyond-mean-field approach with covariant energy density functional