For a maximal ideal $\mathfrak m$ of some anemic Hecke algebra $\mathbb T^S_\xi$ of a similitude group of signature $(1,d-1)$, one can associate a Galois $\overline{\mathbb F}_l$-representation $\overline \rho_{\mathfrak m}$ as well as a Galois $\mathbb T_{\xi,\mathfrak m}^S$-representation $\rho_{\mathfrak m}$. For $l\geq d$, on can also define a monodromy operator $\overline N_{\mathfrak m}$ (resp. $N_{\mathfrak m} \otimes_{\overline{\mathbb F}_l} \overline{\mathbb F}_l$) on $\overline \rho_{\mathfrak m}$ (resp. on $\rho_{\mathfrak m} \otimes_{\overline{\mathbb Z}_l} \overline{\mathbb F}_l$) giving rise to partitions $\underline{\bar d_{\mathfrak m}}$ (resp. $\underline d_{\mathfrak m}$) of $d$. As with Mazur's principle for $GL_2$, analysing the difference between these partitions, we infer informations about the liftings of $\overline \rho_{\mathfrak m}$ in characteristic zero.