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For a central division algebra $D$ of dimension $d^2$ over a finite extension$F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristicprime to $p$, and an irreducible smooth $R$-representation $\pi$ of$G=GL_n(D)$, we show that for small enough compact open pro-$p$ subgroup $K$ of$G$, the restriction of $\pi$ to $K$ is the same as that of a virtualrepresentation $\sum c_\pi(\lambda) Ind_{P_\lambda}^G 1$, where the sum is overpartitions $\lambda$ of $n$ and $P_\lambda$ a parabolic subgroup of $G$associated to $\lambda$. When $K$ is a Moy-Prasad subgroup of $G$ we determinefrom the $c_\pi(\lambda)$ a polynomial $P_{\pi,K}$ of degree $d(\pi)$independent of the choice of $K$, such that for large enough integers $j$ thedimension of the points of $\pi$ fixed under the congruence subgroup $K_j$ of$K$ is $P_{\pi,K}(q^d)$ where $q^d$ is the cardinality of the residue field of$D$.

Representations of $GL_n(D)$ near the identity