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 $\overline N_{\mathfrak m}$ as well as $N_{\widetilde{\mathfrak m}}$ for every prime ideal $\widetilde{\mathfrak m} \subset \mathfrak m$, giving rise to partitions $\underline{\bar d_{\mathfrak m}}$ and $\underline d_{\widetilde{\mathfrak m}}$ of $d$. As with Mazur's principle for $GL_2$, analysing the difference between these partitions, we infer informations about %the set of prime ideals $\widetilde{\mathfrak m} \subset \mathfrak m$, i.e. the liftings of $\overline \rho_{\mathfrak m}$ in characteristic zero known as level lowering problem.