When $F/Q_p$ is a finite extension, the group $G$ of $F$-points of a connected reductive $F$-group has finitely many nilpotent orbits in Lie $G$. Which finiteness property of admissible irreducible representations of $G$ is linked to the finiteness of nilpotent orbits? For complex representations, this is the Howe Harish-Chandra germ expansion of the trace, around the identity. When $G$ is an inner form of $GL_n(F)$ for an integer $n\geq 2$, or $SL_2(F)$, and $\ell\neq p$ is a prime number, I will present a finiteness property of mod $\ell$-representations of $G$ around the identity, linked to the finiteness of nilpotent orbits. This is common work with Guy Henniart.